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• Spherometer - Measure Least Count and radius of a curvature | Labkafe

Jul 23, 2021 / By Soumen Mondal / in Learning Physics Experiments

To determine the radius of curvature of a given spherical surface by a Spherometer.  

Apparatus:  

• Spherometer
• Half Meter Scale
• Convex Lens

A spherometer is a measuring instrument used to measure the radius of curvature of a spherical surface and a very small thickness. 

Figure 3.1 is a schematic diagram of a single disk spherometer. It consists of a central leg OS, which can be raised or lowered through a threaded hole V (nut) at the centre of the frame F. The metallic triangular frame F supported on three legs of equal length A, B and C. The lower tips of the legs form three corners of an equilateral triangle ABC and lie on the periphery of a base circle of known radius, r. The lower tip of the central screw, when lowered to the plane (formed by the tips of legs A, B and C) touches the centre of triangle ABC. A circular scale (disc) D is attached to the screw.  The circular scale may have 50 or 100 divisions engraved on it. A vertical scale P marked in millimetres or half-millimetres, called main scale or pitch scale P is also fixed parallel to the central screw, at one end of the frame F. This scale is kept very close to the rim of disc D but it does not touch the disc D. This scale reads the vertical distance when the central leg moves through the hole V.   

Fig 3.1       

Principle:  

Pitch of a Spherometer  

            The vertical distance moved by the screw S in one complete rotation of the circular Scale/Disc D is called the pitch (p) of the spherometer. To find the pitch, give full rotation to the screw (say 4 times) and note the distance (d) advanced over the pitch scale. 

If the distance d is 4 mm The pitch can be represented as, 

Least Count of the Screw Gauge  

The Least count (LC) is the distance moved by the spherometer screw, when the screw is turned through 1 division on the circular. We are using a spherometer which has 100 divisions (N) on the disc. The least count can be calculated using the formula, 

The formula for the radius of curvature of a spherical surface  

Approach 1:  

From the figure 3.3, O is the centre of the circle. OE = OA = R, radius of the circle. F is the tip of the screw at the same plane with A, B and C. EF = h, AF = a and ∠AFO =   

Therefore, geometrically we can write, 

OA2 = OF2 + FA2 

or, R2    = (R-h)2 + a2 

= R2 -2.R.h + h2 + a2 

∴  R = (h2 + a2 )/2h  

Now, let  l  be the distance between any two legs of the spherometer as shown in figure 3.6, then from geometry we have, a = . Thus the radius of curvature of the spherical surface can be given by, 

               ∴ R = ( 3h2 + l2 )/6h         

Approach 2:  

From the figure 3.4, the circle is passing through A and C.  O is the centre of the circle. OE =R, radius of the circle. F is the tip of the screw at the same plane with A, B and C. CF = h, AF = a ∠EAC = 900. 

∴ CE2 = AE2 + AC2 

or, (2R2) = (AF2 + FE2) + (CF2 + AF2) 

                  = a2 + (2R -h)2 + h2 + a2 

∴ R=  a2/2h+ h/2 

Now, let  l  be the distance between any two legs of the spherometer or the side of the equilateral triangle ABC (Fig. 3.4), then from geometry we have, a = l/√3. Thus the radius of curvature of the spherical surface can be given by, 

R =  a2/2h+ h/2 

or,  R = l2/6h+ h/2  

Diagram:   

Procedure:  

• Find the pitch (p) of the screw and count the total number of divisions (N) in the circular scale.
• Place the spherometer in the plane glass plate. Now rotate the head T anti-clockwise to raise the tip of the central screw S by a certain distance.
• Place the spherometer on the convex surface. Gently rotate T clockwise to bring down the tip of S until it just touches the spherical surface. Use a paper strip and try to pass between the tip of the screw and spherical surface to check if there is no gap between them.
• Record the initial circular scale reading (r1) in table 3.1. Circular scale reading means the divisions engraved on the disc which coincides with the linear scale.
• Place the spherometer on the glass slab without disturbing the initial circular scale reading (c.s.r). Then slowly rotate T clockwise to bring the tip down and touch the glass plate. During this rotation count the number of full rotation (n) of the circular scale. Take the final c.s.r. (r2) when the tip touches the glass plate.
• Repeat step 2 and 5 at least thrice by placing the spherometer at different places.
• Now, place the spherometer on a piece of paper and press it lightly so that an imprint of the three legs is made on the paper. You can do it on your laboratory notebook on the left side white page.
• Measure each side of the triangle AB, BC, and CA formed by the points  (A, B, C).
• Take mean of them. Thus we get l.

Observations:  

Least count of spherometer :  

Total number of divisions is the in circular scale, N = _______ 

One linear scale division, L.S.D.  = ____ mm 

Distance moved by the screw for 4 rotations, d = ________ mm 

Pitch of the screw, p = 4/d = ____mm 

Therefore, Least Count, L.C. = p/N= ______________mm 

Distance between two legs of spherometer:  

               AB = _______ cm, BC = _______cm, CA = _________cm 

∴ l = (AB + BC+ CA)/3 = _______________________cm 

Table 3.1 Table for height (h)  

Mean value of sagitta, h = _________________ mm = ________________cm 

Calculation:  

Radius of curvature of the given convex surface, R = (3h2 + l2 )/6h =…………………………..cm  

or, Radius of curvature of the given convex surface ,  R = l2/6h+ h/2 =……………………………   cm  

Precautions:  

• The screw should move freely without friction.
• The screw should be rotated in one direction to get any reading. Otherwise back-lash error will be introduced.
• The circular should not be rotated any more, even slightly, when it touches a surface.
• The linear scale is not used to take readings and h is calculated by taking the difference of two circular scale readings. Hence we do not need to find the zero error of the instrument.

Reference:  

• http://www.ncert.nic.in/

Your may checkout our blog on  SCREW GAUGE & LEAST COUNT    

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To determine the radius of curvature of a given spherical surface by a spherometer

To determine the radius of curvature of a given spherical surface by a spherometer.

Apparatus and material required

A spherometer, a spherical surface such as a watch glass or a convex mirror and a plane glass plate of about 6 cm x 6 cm size.

Description of Apparatus

A spherometer consists of a metallic triangular frame F supported on three legs of equal length A, B and C (Fig. E 3.1). The lower tips of the legs form three corners of an equilateral triangle ABC and lie on the periphery of a base circle of known radius, r. The spherometer also consists of a central leg OS (an accurately cut screw), which can be raised or lowered through a threaded hole V (nut) at the centre of the frame F. The lower tip of the central screw, when lowered to the plane (formed by the tips of legs A, B and C) touches the centre of triangle ABC. The central screw also carries a circular disc D at its top having a circular scale divided into 100 or 200 equal parts. A small vertical scale P marked in millimetres or half-millimetres, called main scale is also fixed parallel to the central screw, at one end of the frame F. This scale P is kept very close to the rim of disc D but it does not touch the disc D. This scale reads the vertical distance which the central leg moves through the hole V. This scale is also known as pitch scale.

Terms and Definitions

Pitch: It is the vertical distance moved by the central screw in one complete rotation of the circular disc scale. Commonly used spherometers in school laboratories have graduations in millimetres on pitch scale and may have100 equal divisions on circular disc scale. In one rotation of the circular scale, the central screw advances or recedes by 1 mm. Thus, the pitch of the screw is 1 mm.

Least Count: Least count of a spherometer is the distance moved by the spherometer screw when it is turned through one division on the circular scale, i.e.,

Least count of the spherometer =Pitchof thespherometerscrew /Numberof divisions on the circular scale

The least count of commonly used spherometers is 0.01 mm. However, some spherometers have least count as small as 0.005 mm or 0.001 mm.

Formula for The Radius of Curvature of A Spherical Surface

Let the circle AOBXZY (Fig. E 3.2) represent the vertical section of sphere of radius R with E as its centre (The given spherical surface is a part of this sphere). Length OZ is the diameter (= 2R ) of this vertical section, which bisects the chord AB. Points A and B are the positions of the two spherometer legs on the given spherical surface. The position of the third spherometer leg is not shown in Fig. E 3.2. The point O is the point of contact of the tip of central screw with the spherical surface. Fig. E 3.3 shows the base circle and equilateral triangle ABC formed by the tips of the three spherometer legs. From this figure, it can be noted that the point M is not only the mid point of line AB but it is the centre of base circle and centre of the equilateral triangle ABC formed by the lower tips of the legs of the spherometer (Fig. E 3.1). In Fig. E 3.2 the distance OM is the height of central screw above the plane of the circular section ABC when its lower

tip just touches the spherical surface. This distance OM is also called sagitta. Let this be h. It is known that if two chords of a circle, such as AB and OZ, intersect at a point M then the areas of the rectangles described by the two parts of chords are equal. Then

AM.MB = OM.MZ

(AM) 2 = OM (OZ - OM) as AM = MB

Let EZ (= OZ/2) = R, the radius of curvature of the given spherical surface and AM = r, the radius of base circle of the spherometer.

r 2 = h (2R - h)

Thus, R = r 2 /2h + h/2

Now, let l be the distance between any two legs of the spherometer or the side of the equilateral triangle ABC (Fig. E 3.3), then from geometry we have

Thus, r = 1/√ 3 , the radius of curvature (R) of the given spherical surface can be given by

R= ι 2 /6h + h/2

• Note the value of one division on pitch scale of the given spherometer.
• Note the number of divisions on circular scale.
• Determine the pitch and least count (L.C.) of the spherometer. Place the given flat glass plate on a horizontal plane and keep the spherometer on it so that its three legs rest on the plate.
• Place the spherometer on a sheet of paper (or on a page in practical note book) and press it lightly and take the impressions of the tips of its three legs. Join the three impressions to make an equilateral triangle ABC and measure all the sides of Δ ABC. Calculate the mean distance between two spherometer legs, l.
• In the determination of radius of curvature R of the given spherical surface, the term ι 2 is used (see formula used). Therefore, great care must be taken in the measurement of length, ι.
• Place the given spherical surface on the plane glass plate and then place the spherometer on it by raising or lowering the central screw sufficiently upwards or downwards so that the three spherometer legs may rest on the spherical surface (Fig. E 3.4).
• Rotate the central screw till it gently touches the spherical surface. To be sure that the screw touches the surface one can observe its image formed due to reflection from the surface beneath it.
• Take the spherometer reading h1 by taking the reading of the pitch scale. Also read the divisions of the circular scale that is in line with the pitch scale. Record the readings in Table E 3.1.
• Remove the spherical surface and place the spherometer on plane glass plate. Turn the central screw till its tip gently touches the glass plate. Take the spherometer reading h 2 and record it in Table E 3.1. The difference between h 1 and h 2 is equal to the value of sagitta (h).
• Repeat steps (5) to (8) three more times by rotating the spherical surface leaving its centre undisturbed. Find the mean value of h.

Observations

A. Pitch of the screw:

• Value of smallest division on the vertical pitch scale = ... mm
• Distance q moved by the screw for p complete rotations of the circular disc = ... mm
• Pitch of the screw ( = q / p ) = ... mm

Least Count (L.C.) of the spherometer:

• Total no. of divisions on the circular scale (N ) = ...
• Least count (L.C.) of the spherometer
• = Pitchof thespherometerscrew /Numberof divisionsonthecircular scale
• L.C. = Pitchof thescrew /N = ... cm

Determination of length l (from equilateral triangle ABC)

• Distance AB = ... cm
• Distance BC = ... cm
• Distance CA = ... cm
• Mean ι = AB + BC+ CA /3= ... cm

Table E 3.1 Measurement of sagitta h

Mean h = ... cm

Calculation

Using the values of l and h, calculate the radius of curvature R from the formula:

R = ι 2 /6h + h/2;

the term h/2 may safely be dropped in case of surfaces of large radii of curvature (In this situation error in ι 2 /6h is of the order of h/2.)

The radius of curvature R of the given spherical surface is ... cm.

Precautions

• The screw may have friction.
• Spherometer may have backlash error.

Sources of Error

• Parallax error while reading the pitch scale corresponding to the level of the circular scale
• Backlash error of the spherometer.
• on-uniformity of the divisions in the circular scale.
• While setting the spherometer, screw may or may not be touching the horizontal plane surface or the spherical surface.

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Measuring with Spherometer: A Virtual Lab Experiment

What is the Radius of the Curvature of the Spherical Surface? 

Spherical surfaces are the part of the sphere which is used to form the image as per the requirement of an object using the principle of reflection of light. There are two types of spherical surfaces: convex and concave.

The linear distance between the pole and centre of curvature is called the radius of curvature. The centre of the spherical surface is called the pole, whereas the centre of the sphere (from which the spherical surface is cut ) is called the centre of curvature. When the radius of curvature becomes infinite, the spherical mirror behaves as a plane mirror. The radius of curvature lies on the principal axis of the spherical surface.

Diagram of Spherometer

Given below is the labeled diagram of a Spherometer.

How to read Spherometer?

The following is the procedure for using a spherometer:

• First, place the instrument on the perfect plane surface, so the central leg is screwed down slowly until it touches the surface. When the central leg touches the surface, the instrument rounds on the central leg as the centre.
• Remove the spherometer from the surface to take the reading from the micrometre screw. If the instrument works fine, the reading should be 0-0. However, there is always a slight error in the instrument, which could be either a positive or negative error.
• Take the instrument off the plane and draw the central leg back.
• Let’s consider measuring the sphere’s radius from the convex side.
• Now read the scale and screw-head. If the reading is 2.0 and 0.155, then the total reading is 2.155.
• If the reading is below the zero lines, then the reading should be added to the zero error. If the reading is above the zero lines, then the reading should be subtracted from the zero error.
• To measure the length between the two legs, place the instrument on the plain card and measure the length using a meter scale.
• The radius of curvature can be calculated using the following equation:

What is the Least Count of Spherometer?

The least count (L.C) can be calculated using the relation,

Pitch of a Spherometer: The pitch is defined as the distance covered by the circular disc in one complete rotation along the main scale. Therefore, the pitch of a spherometer is given as 1 mm = 0.1 cm.

Number of circular divisions = 100

What is Zero Error in Spherometer?

A zero error is an error in your readings determined when the true value of what you’re measuring is zero, but the instrument reads a non-zero value. 

A spherometer does not have a zero error because the result obtained is by taking the difference between the final and initial reading.

Applications of Spherometer

The primary application of a spherometer is to measure the radii of curvature of spherical surfaces such as optical lenses, spherical mirrors, and balls. These small, high-precision optical test instruments are also used to measure the thickness of microscope slides or the depth of slide depressions.

Solved Examples for Spherometer

Ex-1. A student measures the height h of a convex mirror using a spherometer. The legs of the spherometer are 4 cm apart, and there are ten divisions per cm on its linear scale, and the circular scale has 50 divisions. The student takes two as linear scale division and 40 as circular scale division. What is the radius of curvature of the convex mirror?   

The values of I and h are 4.0 cm and 0.065 cm, respectively, where ‘I’ is measured by a meter scale and h by a spherometer. Find the relative error in the measurement of R.

Spherometer Experiment

Experiment Title – Use of Spherometer to Find Radius of Curvature  

Experiment Description – A spherometer is a precision instrument that measures very small lengths. Let’s determine the radius of curvature of a given spherical surface using a spherometer.

Aim of Experiment – To determine the radius of curvature of a given spherical surface by a spherometer.

Material Required – A spherometer, a convex glass surface, a plane glass plate, a pencil, a measuring scale, a paper sheet and a small piece of paper.

Procedure – 

• Observe the given spherometer and note the value of one division of its pitch scale.
• Observe the circular scale and note the number of divisions on it.
• Determine the least count (L.C.) and pitch of the spherometer. Place the given flat glass plate on a horizontal plane and the spherometer on it so that its three legs rest on the plate.
• Take a sheet of paper, place the spherometer on it, and press it gently to take the impressions of the tips of the three legs. Make an equilateral triangle ABC by joining the three impressions and measuring all the sides of the ΔABC. Determine the mean distance between two spherometer legs, l.

Take great care in measuring the length l as the term l 2 is used to determine curvature R of the given spherical surface.

• Place the given spherical surface on the plane glass plate and then place the spherometer on it by raising the central screw sufficiently upwards so that the three legs of the spherometer rest on the spherical surface, as shown in the figure below. 

• Rotate the central screw till its lower tip gently touches the spherical surface. Observe the image of the screw formed due to the reflection from the surface below to make sure that the screw touches the surface.
• Observe the reading of the pitch scale and the divisions of the circular scale that is in line with the pitch scale to take the spherometer reading h 1 . Record the observations in the observation table. 
• Remove the spherical surface and place the spherometer on the plane glass plate. Turn the central screw till its lower tip gently touches the glass plate. Again, take the spherometer reading h 2 and record it in the observation table. The difference between h 1 and h 2 equals the value of sagitta (h).
• Repeat steps (5) to (8) three more times by rotating the spherical surface without disturbing its centre. Find the mean value of h.

Precautions – 

• The screw of the spherometer may have friction. 
• The spherometer may have a backlash error. 

FAQs on Spherometer

It works on the principle of a micrometre screw.

A spherometer has three legs so that it forms an equilateral triangle. The three legs of the spherometer are used for measuring positively and negatively curved surfaces.

The pitch is the distance covered by the circular disc in one complete rotation along the main scale. Therefore, the pitch of a spherometer is given as 1 mm = 0.1 cm.

The accuracy of the spherometer can be increased by decreasing the pitch or increasing the number of divisions of the circular scale. The smaller the least count, the more the accuracy of an instrument and vice versa.

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• Units Of Measurement
• Standard Measurement Units

Spherometer

What is spherometer.

A spherometer is defined as

An instrument that is used for precise measurements of the radius of curvature of either sphere or a curved surface.

The first spherometer was invented by Robert-Aglae Cauchoix who was a French optician in 1810. These were primarily manufactured for the use of opticians in grinding lenses. Astronomers also used this instrument in grinding lenses and curved mirrors .

Spherometer Diagram

Spherometer Working Principle

The working principle of a spherometer is based on the micrometer screw. It is used for measuring with a small thickness of flat materials such as glass or for measuring the radius of curvature of a spherical surface.

Parts of Spherometer

A spherometer generally consists of a base circle of three outer legs, a central leg and a reading device.

• A spherometer consists of a base circle of three outer legs, which is also known as the radius of the base circle, a ring with a known radius of the base circle.
• The outer legs of the spherometer can be adjusted according to the inner holes. This is done to accommodate smaller surfaces.
• The central leg can be moved in an upward and downward direction.
• For taking the measurements, the reading device on the central leg should be moved.

Principles of Operation

If R is the radius of spherical material, then the mean length between two outer legs can be determined by using the formula:

Where h is the sagittal measure.

The spherical radius R can be determined by a different spherometer without legs and with circle cup and dial gauge, D is given by the formula:

Least Count of Spherometer

Number of divisions on the circular scale = 100

Distance moved by the screw in 10 complete rotations = 10 mm

Pitch = Distance moved/number of complete rotations

Least count = Pitch/number of divisions on the head scale = 1/100 = 0.01 mm

How to use a Spherometer?

The following is the procedure to use spherometer:

• The instrument is first placed on the perfect plane surface such that the middle foot is screwed down slowly till it touches the surface. When the middle foot touches the surface, the instrument turns rounds on the middle foot as the centre.
• The spherometer is then carefully removed from the surface to take the reading from the micrometre screw. If the instrument is working fine, then the reading should be 0-0. However, there is always a slight error in the instrument which could be either a positive or negative error.
• Take the instrument off the plane and draw the middle foot back.
• Let’s consider that we are measuring the radius of the sphere from the convex side.
• Now read the scale and screw-head. If the reading is 2.0 and 0.155, then the total reading is 2.155.
• If the reading is below the zero lines, then the reading should be added to the zero error. If the reading is above the zero lines then the reading should be subtracted from the zero error.
• To measure the length between the two legs, the instrument should be placed on the plain card and using a meter scale the length should be measured.
• Now, calculate the radius of curvature using the following equation:

Stay tuned with BYJU’S to learn more about other concepts of Physics.

Frequently Asked Questions – FAQs

Why does a spherometer have three legs.

A spherometer has three legs so that it forms an equilateral triangle. The three legs of the spherometer are used for measuring both positively and negatively curved surfaces.

How to find the zero error in a spherometer?

A spherometer does not have a zero error because the result obtained is by taking the difference between the final and initial reading.

Why is a spherometer so called?

A spherometer is so-called because it is used for measuring the radii of curvature of spherical surfaces.

What is the pitch of spherometer?

The pitch is defined as the distance covered by the circular disc in one complete rotation along the main scale. Therefore, the pitch of a spherometer is given as 1 mm = 0.1 cm.

What is zero error in a spherometer?

The zero error in a spherometer is equal to the reading on the plane glass sheet.

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• Determine Radius of Curvature of a Given Spherical Surface by a Spherometer

An Introduction

Class 11 students are expected to know the exact spherometer experiment procedure. You will need to determine the radius of curvature of given spherical surfaces using this device. However, before proceeding to learn about the process of doing the same, one should understand spherometers in detail.

What is a Spherometer?

A spherometer is one of the vital scientific devices that measure the radius of curvature for any spherical surface precisely. Initially, opticians used these devices to create and determine powered lenses. 

We come across various instruments that are used in a lab for the measurements of different things but when we have to measure the radius of curvature of either a sphere or a curved surface with its precise measurements, then we use an instrument called a spherometer. Now if you look back at the history of the spiral meter we see that it was invented by Robert-Aglaé Cauchoix, his profession was that of an optician in the year 1810. Robert mainly manufactured the spherometer for the use of opticians in grinding lenses. Other than using it for grinding lenses or in the physics lab, sphere meters are used by astronomers for grinding lenses and curved mirrors. Accordingly, spherometers can have various other uses as well.

Now before we go ahead with the understanding and definition of the spherometer we will see the working principle of the device. The working principle of a spherometer is based on a micrometer screw which is used for measuring a small thickness of flat material such as gas or can be used for measuring the radius of curvature of a spherical surface.

Normally a spherometer can be described as a device consisting of a base of a  circle of three of the leg, central leg and a reading device. The Circle of three or three legs is also known as the radius of the base Circle and the land along with it is known as the radius of the base circle, the outer legs which are given can be adjusted accordingly depending on the inner holes, this procedure is mainly done to accommodate smaller surfaces. The central leg of the spherometer can be moved in an upward and downward direction accordingly, this can be called a flexible method of drawing lines or using it with any other measurements. For taking the measurements any device on the reading device should be moved accordingly.

How to use a Spherometer?

A spherometer is a very common device in labs, opticians, and other physics-related settings but the main thing that is necessary is to know and understand how to use a spherometer. There is a certain set of procedures that are included while using a speedometer: 

After holding the instrument, it is first placed on the perfect plane surface in a manner that the middle foot is screwed down slowly till it touches the surface below. After the middle foot touches the surface, the instrument turns around on the middle foot as the center, now the center has a point from which we can equally draw shapes accordingly or measure.

After the surface is made the spherometer is then carefully removed from the surface to take the readings from the micrometer screw. The instrument should show the reading 0-0 in normal cases, this reading comes when the instrument is working fine, any other reading might lead to errors. If there is any sort of slight error in the instrument it could be either a negative or positive error.

Now, when we measure the instruments reading, we take the instrument of the plane and let the middle foot back.

When we are measuring we come across a reading below the zero line so if we see that reading it should be added to the zero error. If the reading above the zero lines is indicated then the reading should be subtracted from the zero error in order to make it balanced.

To measure the length between the two legs, the instrument should be placed on the plane surface or a playing card while using a meter scale so that equal length is measured

To measure the radius of curvature using a spherometer, one must know its various parts. The device has a screw with a moving nut in the middle of a frame with three small legs to support it upright. The table legs, along with the screw, have tapered points to help them rest on a specific surface.

Additionally, a spherometer’s least count can differ from one device to another and each time that we may use it to determine the radius of curvature, we still need to calculate this acquired count again. 

Define Spherometer Least Count

The smallest value that a spherometer can measure is known as its least count. The formula for determining the least count is as follows – 

Least count = Pitch/Number of divisions on its head scale

Typically, the least count is always 0.01 mm. 

Experiment to Find Radius of Curvature using Spherometer

Now that you know some of this measuring device’s basics, let us learn more about this experiment in general.

To find the radius of curvature using spherometer of a spherical surface

Apparatus Necessary

Plane mirror, spherometer and convex surface

Table Format for Noting Experiment Data

Complete Procedure for the Experiment

Step 1: Raise the central screw of this device and use a paper to track the position of a spherometer’s three legs. Join these three points on the paper and mark them A,B and C. 

Step 2: Measure the minute distance between the three points. Note the three distances (AB, BC and AC) on a sheet of paper.

Step 3: Determine the value of one pitch (or one vertical division).

Step 4: Record the least count of your spherometer.

Step 5: Raise the screw upwards to prepare for the measurement.

Step 6: Place this spherometer on the spherical surface in such a manner that all three legs are resting on the object.

Step 7: Start turning the screw so that it barely touches this convex surface. 

Step 8: Take the reading of both the vertical scale and the disc scale in such a position. This will act as your reference point.

Step 9: Now place this spherometer on a plane glass slab.

Step 10: Move the screw downwards and count the number of complete rotations for the disc (n 1 ).

Step 11: Continue moving until the screw tip touches the glass slab. 

Step 12: Note the reading (b) on this circular scale in relation to its vertical scale.

Step 13: Note the circular divisions for its last incomplete rotation.

Step 14: Complete steps 6 to 13, thrice. Note readings each time in the tabular format mentioned above.

Observations

Mean Value of AB, BC and AC

Mean value or l = \[\frac {AB + BC + AC}{3}\]

Mean Value of h

h = \[\frac {h_1 + h_2 + h_3}{3}\] mm (Convert into cm)

Calculating Radius of Curvature of Convex Lens using Spherometer Readings

Radius R = \[\frac {I^2}{6h}\] + \[\frac {h}{2}\]   cm

Vedantu’s interactive classes can help you understand more about spherometer readings, the radius of curvature and more. Experienced teachers are at your disposal whenever you need your doubts cleared.  It helps a child strengthen his or her basic concepts when understood in a processed and elaborate manner and Vedantu uses point-to-point examples and explanations from the given terms so that it becomes easier to understand complex terms as well.  you can also download our Vedantu app for better access to these study materials and online interactive sessions.

FAQs on Determine Radius of Curvature of a Given Spherical Surface by a Spherometer

1. What is the Value of R for a Plane Surface?

Radius of curvature R only exists for spherical surfaces. Therefore, R for a plane surface will always be zero.

2. What is this Formula to Calculate Radius of Curvature Using a Spherometer?

Radius R =   \[\frac {1^2}{6h}\]    + \[\frac {h}{2}\] cm  is the formula which is used to calculate the radius of curvature of a spherometer.

3. What is the Formula to Measure the Least Count of Spherometers?

You can determine the least count of a spherometer by dividing the pitch of the device by the number of divisions on a circular scale.

4. What is the use of a spherometer?

A spherometer is an instrument used for the precise measurement of the radius of curvature of a sphere or a curved surface. 

5. What is the least count of a spherometer?

The least count can be defined as the distance moved or covered by the crew of the spherometer when turned through 1 division on the circular loop. The least count can be calculated using the formula, The formula for the radius of curvature of a spherical surface. The least count of the spherometer can be measured by dividing the pitch of the spherometer screw by the number of divisions on the circular scale.

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Spherometer

The spherometer is used to measure radius of curvature of spherical surfaces (e.g., spherical lens and mirror). Its working principle is same as that of a screw gauge .

Construction of a spherometer

A spherometer has three fixed legs and an adjustable central legs. The central leg is has a micrometer head on its top. The micrometer has a main (vertical) scale and a circular scale.

The pitch ($p$) of a spherometer is the distance moved in one complete revolution of the circular scale. The least count (LC) of a spherometer of pitch $p$ and having $N$ divisions on the circular scale is given by \begin{align} \text{Least count}=\frac{p}{N}. \end{align} Usually, p = 1 mm, N=100, and LC = 0.01 mm.

Derivation of spherometer formula

The tips of three fixed legs touches the spherical surface to be measured. These tips forms a plane. They lies on an equilateral triangle of side $l$. The distance ($l$) between tips of fixed legs is measured with the help of a scale.

Initially, the spherometer is placed on a flat surface. Its central legs is adjusted (moved) so that tip of the central leg lies at the centroid of the equilateral triangle formed by fixed legs. In this condition, the distance between the tip of central leg and any fixed leg is $r$. From geometry, \begin{align} r\cos30=\frac{l}{2} \end{align} which gives \begin{align} r=\frac{l}{\sqrt{3}} \end{align}

Now, spheromoeter is placed on the spherical surface of radius of curvature $R$. All fixed legs shall touch the surface. The central leg is adjusted so that its tip also touches the spherical surface. The distance $h$ between the plane containing tips of three fixed legs and the tip of central leg is measured with the help of main scale and the circular scale. The central leg can be moved up or down from the zero mark on the vertical scale.

From Pythagoras triangle \begin{align} R^2 &=r^2+(R-h)^2 \end{align} Simplify and substitute $r=l/\sqrt{3}$ to get \begin{align} R&=\frac{l^2}{6h}+\frac{h}{2} \end{align}

Problems on spherometer

Problem: The pitch of a spherometer is 1 mm and there are 100 divisions on its disc. It reads 3 divisions on the circular scale above zero when it is placed on a plane glass plate. When it rests on a convex surface, it reads 2 mm and 63 divisions on a circular scale. If the distance between its outer legs is 4 cm, the radius $R$ of curvature of the convex surface is

Solution: The least count of the spherometer is \begin{align} \text{LC}&=\frac{p}{N}=\frac{1}{100}=0.01\;\text{mm} \end{align} The zero error of the spherometer is its measurement on the plane surface, which is 3 divisions on the circular scale. Thus, zero error is \begin{align} \text{Zero error}=3\times\text{LC}=0.03\;\text{mm}. \end{align} The measured value of $h_m$ is 2 divisions on the main scale and 63 divisions on the circular scale i.e., \begin{align} h_m&=2+63\times\text{LC}=2.63\;\text{mm} \end{align} The zero correction gives \begin{align} h &=h_m-\text{Zero error}\\ &=2.63-0.03 \\ &=2.60\;\text{mm}=0.26\;\text{cm} \end{align} The distance between fixed legs is $l=4$ cm. Apply the spherometer formula to get the radius of curvature of the convex surface \begin{align} R &=\frac{l^2}{6h}+\frac{h}{2}\\ &=\frac{4^2}{6(0.26)}+\frac{0.26}{2} \\ &=10.4\;\text{cm}. \end{align}

Question: How will you find the focal length of lens by using a spherometer?

Answer: The focal length of a lens is given by the Lens maker's formula \begin{align} \frac{1}{f}=(\mu-1)\left[\frac{1}{R_1}-\frac{1}{R_2}\right] \end{align} The spherometer is used to measure the radius of curvatures $R_1$ and $R_2$ of two surfaces of the lens. The refractive index $\mu$ of the lens material is known.

Question: What do you mean by radius of curvature of a surface?

Answer: It is the radius of the sphare from which the surface is cut. You can cut a glass sphere of radius $R$ to get a plano-convex lens. The radius of curvature of the curved surface of the lens is $R$ and that of the plane surface is infinity.

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Determining the Radius of Curvature of a Given Spherical Surface by Spherometer, Steps, Sample Questions

Collegedunia Team

Content Curator

Radius of curvature is the reciprocal of a curve at a mentioned point. A spherometer is used to determine the radius of curvature. The aim of a spherometer is to measure the curvature of a spherical surface so it can be convex or concave by using a spherometer.

Key terms: Radius of Curvature, Spherometer, Least count , Pitch, Watch Glass, Convex Surface, Concave Surface

Radius of Curvature

[Click Here for Sample Questions]

Radius of curvature is-

R= l 2 /6h+h/2

Where h= height up to which the spherical surface has bent.

l= leg length of the spherometer (Leg length is the average length of the leg calculated using a scale.) 

r= radius of the curvature

Also Read:  Surface Energy

Few Common Terms to Understand

Before getting into the depths of the experiment few terms are enlisted herein:-

• Least count- The least count means the distance measured by the spherometer. The Least count Formula is:

[Pitch/No of Division on the circular scale]

• Pitch: Pitch is the distance covered between two points of the screw. It means by giving one rotation, only one thread is advanced in the screw. So, if only one thread is advanced, then the distance covered in the main scale is the pitch. The division covered in the circular scale is 100 (0 to 90).

Now, Let's take an example to understand the process to calculate Radius of Curvature.

Assume h 1 as 1, h 2 as 2, and h 3 as 3, and the length of the three sides of the triangular surface is 3.2, 3.2, and 3.1. Calculate the Radius of the spherical surface using a tabular format.

The solution,

• Mean of l= [AB+BC+CA/3]
• Calculate the least count
• As 5 rotation to the circular scale is given, it shows the effect in the main scale is i.e. it completed around 5 millimeters in the main scale.
• By giving one millimeter= 5/5= 1 millimeter= 0.1 cm which is the pitch or the distance covered.
• Least count = [Pitch/No of Division on the circular scale]

Or, 0.1/100= 0.001cm

The measurement should be taken using a spherometer.

• Sagitta Table (h)

• If the value of h was not given in the question, then find out every value of h.
• Mean of h= [h 1 +h 2 +h 3 /3]= [1+2+3/3]= 2mm
• Mean of l= [(3.2+3.2+3.1)/3] = 3.16cm

Therefore, R= {(3.1) 2 /6*2}+{2/2}= 1.80 cm.

Also Read:  Viscosity

What is a Spherometer?

It is a simple three-legged scientific device that determines the radius of curvature. The spherometer has a big middle leg. The middle big leg has a rotating screw that is circulated to coincide the edge with zero.

Spherometer

Experiment to Determine the Procedure of Calculating Radius of Curvature Using Spherometer

The radius of the curvature can be determined simply through a small experiment. The apparatus required is the spherometer and the watch glass. The glass is required because it has a convex or a concave spherical surface. 

Steps for Experiment

• Calculate the least count of the spherometer. 
• Use the circular and vertical scale together in a spherometer. 
• Give 5 rotations to the circular scale and mark the reading from the circular scale. Edge is the reference point. (Give rotation in such a manner that 53 points are completed)
• Circulate the screw and give 5 rotations.
• You can give clockwise or anti-clockwise rotation as the only thing that gets calculated is the distance that is covered in the main scale.

Also Read:  Venturi-meter

Observation of the Experiment Presented in Tabular Format

Using the above data the below-mentioned table is presented to determine the observations.

The process of taking the observations is as follows: 

• Take a watch glass and the procedure of convex and concave spherical surfaces are different. 
• For such a procedure, one must consider at least 10 observations to minimize the error. Herein, only one observation is considered to explain the procedure. Similarly, 9 more observations to consider to complete the process. 
• Keep the watch-glass on the convex surface side and then take the spherometer and keep it above the watch-glass.
• Give rotation to the circular scale till the middle leg of the spherometer touches the top surface of the convex spherical surface of the watch-glass. 
• Further, to understand whether the same is touching the glass, take a small piece of paper to check the same. 
• If all the three legs start rotating, it means the middle leg has touched the above surface. 
• If the small paper passes from between the surface of the glass and the middle leg of the spherometer, it means the middle leg has not touched the glass yet. So, place it properly and give back rotations. 
• Give the exact position where the middle leg touches the surface. 
• Then, take the reading from the circular scale. And the edge of the main scale is the reference. 
• Now, suppose the main scale is coinciding with 21 points of the circular scale, then 21 is the initial circular scale reading. 
• Further, observe NCR. Take the spherometer and keep it on a flat surface. 
• A small gap lies between the middle leg of the spherometer and the flat surface. You can use a small piece of paper to check the same. 
• Now, rotate the circular scale till the middle leg touches the upper surface of the flat area. 
• Previously, it was noted that the reference point was 21. So, taking that as the base, rotate the spherometer till the middle leg touches the surface. 
• After the number of rotations gets completed and the middle leg touches the surface, the exact point is determined. (where the paper does not pass from between the middle leg and the surface of the flat).
• Then, note the reading from the circular scale edge. Suppose the reading is 4.
• Now, INF calculation

 if I > F, then INF is (I - F)

 Where I = Initial Circular scale reading

F= Final Circular Scale Reading

And if I < F, then add the number of divisions in the circular scale i.e. 100 and I, and deduct F 

i.e. [(100+I)-F]

• PSR= Pitch* Number of Complete rotation
• CSR= INF * List Counts

Therefore, in the above table,

• ICSR (assumed reading)= 21
• NCR (assumed reading)= 1
• FCSR (assumed reading)= 4
• INF equals= (21-4)= 17
• PSR= (0.1cm*1)= 0.1 cm

Where, pitch (calculated above) is 0.1

CSR= (17 * 0.001cm)= 0.17cm

Where, 17 is the INF 

0.001 is the least count (calculated above)

• Total= (O.1+0.17)= 0.117cm
• Mean is also calculated as at least 10 observations are taken so that less number of errors of minimum error is committed. 

In such a manner the measurement of the quantity (i.e. the height up to which the spherometer was raised.) 

4. Observation in a Concave Spherical Surface

For a concave spherical surface, the same process will be followed but in a reverse manner. 

Reverse Procedure:

• First, take the Spherometer and place it on a plain surface. 
• Give rotation to the circular scale till it touches the surface. 
• Check the difference gap with a piece of paper in a similar manner followed in a convex surface and adjust it to get the edge reading.
• Then, put the spherometer in its current position on the glass watch and observe the gap. 
• Adjust the rotations in such a way so that the gap diminishes and the middle leg touches the glass surface. 
• Simultaneously, note the number of complete rotations and the FCSR.

Also Read:  Bernoulli’s principle

Things to Remember

• Always use a small piece of paper to determine the gap between the middle leg of the spherometer and the surface.
• Always use at least 10 figures to determine the observations.
• Rotate the screw smoothly. If any obstacle while rotating, then remove such obstruction.
• Select glass watches of the same sizes.

Also Read:  Surface tension

Sample Questions

Ques. How can the Accuracy of the Spherometer increase? (3 marks)

Ans. The accuracy of the Spherometer depends on the size of the least count. If the least is small, the chances of accuracy are more. If the divisions of the circular scale are adjusted, then the accuracy level can be adjusted too. 

Ques. What is the Lens Maker Formula? (3 marks)

Ans. The focal length of the glass is determined by the below-mentioned formula. 

1/f= (n 2 /n 1 -1)(1/r 1 -1/r 2 )

Ques. How does the error arise while calculating the least count? Note any 3 of them. (3 marks)

Ans. the error can arise due to the following reasons:

• Friction and obstruction in the screw while rotating.
• Back-lash error in the spherometer.
• The dimensions of the glass are of different sizes.

Ques. Why was the height of the spherometer measured? (3 marks)

Ans. To determine the extent the height was curved in the glass is measured using a spherometer. And the combinations of more than 10 are used to determine the correct results using the average of the observations or mean. Mean is calculated so that fewer number errors of minimum errors are committed.

Ques. Define the principle of the Spherometer. (3 marks)

Ans. The principle of micrometer screw is applied on the Spherometer. The breadth of the glass is determined from the same. The readings are taken from the edge of the watch glass or any other glass material to determine the radius of curvature of the spherical surface. 

Ques. Explain zero error in Spherometer. (5 marks)

Ans. when the readings are equal in the plain surface, then the zero error possibilities arises. The Spherometer will never have zero error. The observation is the difference between the initial reading and the final reading. So, the difference must be there. If there is similar reading, then perform the procedure again to determine the difference. The difference helps in calculating the other figures which will finally lead to the final measurement of the radius of curvature. 

Ques. Define Backlash Error. (3 marks)

Ans. When the spherometer is in motion the backlash error arises. The spherometer is moved back and forth sometimes which creates a gap between the gear. When the mechanism loses the motion, the gap in the gear is created that finally creates a backlash error. 

Ques. Explain the Parts of the Spherometer in detail. (5 marks)

• It has three outer legs which is the radius of the spherometer. The legs are adjusted accordingly. The smaller surface is adjusted with the three outer legs.
• It has a middle leg that is moved in different directions to calculate the radius of curvature.
• It has a circular scale placed horizontally and the main scale that is placed vertically. When the circular scale coincides with the main scale, determine the radius measurement of the curvature by noting each reading. 
• There is a screw on the top of the middle leg by means of which the rotations are given effect.
• The middle leg is adjusted until the three outer legs touch the desired surface and the readings on the main scale are taken.

Also Read: 

CBSE CLASS XII Related Questions

1. a tank is filled with water to a height of 12.5cm. the apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4cm. what is the refractive index of water if water is replaced by a liquid of refractive index 1.63 up to the same height, by what distance would the microscope have to be moved to focus on the needle again, 2. (a) at what distance should the lens be held from the card sheet in exercise 9.22 in order to view the squares distinctly with the maximum possible magnifying power (b) what is the magnification in this case (c) is the magnification equal to the magnifying power in this case explain., 3. light of wavelength 488nm is produced by an argon laser which is used in the photoelectric effect. when light from this spectral line is incident on the emitter, the stopping (cut-off) potential of photoelectrons is 0.38v. find the work function of the material from which the emitter is made., 4. a capillary tube of radius r is dipped inside a large vessel of water. the mass of water raised above water level is m. if the radius of capillary is doubled, the mass of water inside capillary will be.

\(\frac M4\)

5. A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is

6. a constant power is supplied to a rotating disc. the relationship between the angular velocity $\omega$ of the disc and number of rotations (n) made by the disc is governed by.

• $\omega\propto\,n^{\frac{1}{3}}$
• $\omega\propto\,n^{\frac{2}{3}}$
• $\omega\propto\,n^{\frac{3}{2}}$
• $\omega\propto\,n^2$

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Science Practicals 11 & 12

Search this blog, class 11 physics practical reading to measure radius of curvature of a given spherical (convex) surface by a spherometer., apparatus required.

Observations

Precautions

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Determination of Radius of Curvature of a Spherical Surface by Spherometer

Experiment: Determination of the radius of curvature of a spherical surface by a spherometer.

Theory: Radius of the curvature which is the portion of a sphere is called the radius of curvature of the spherical surface. It is expressed as R.

Radius of curvature, 6 R = (d 2 /6h + h/2)

Here, d = Average distance of the three legs of the spherometer;

And h = height or depth of the spherical surface from the surface of the three legs.

Reading of the spherometer = Main scale reading + circular scale reading x least count.

The distance from the center of a round or sphere to its surface is its radius. For other curved lines or surfaces, the radius of curvature at a given point is the radius of a circle that scientifically best fits the curve at that point.

Apparatus: (1) spherometer, (2) plane plate or a big size plane glass slab, (3) spherical surface, (4) meter scale, (5) convex surface (it may be an unpolished convex mirror), (6) mirror, etc.

Spherometer ant its parts

A Spherometer is an instrument for measuring the curvature of a surface. For example, it can be used to determine the width of a microscope slide or the depth of depression in a slide. The curvature of a ball can also be measured by using a Spherometer.

A spherometer consists of a metallic tripod structure supported on three fixed legs of equivalent lengths. A screw passes through the center of the tripod frame, parallel to the three legs. A large circular disc graduated with 100 equal parts is attached to the top of the screw.

(1) Find out the pitch and least count of the spherometer by knowing the total number of divisions of the circular scale and value of the smallest division of the spherometer.

(2) Instrumental error, if there is any, is also found out.

(3) The spherometer is placed on a plane plate. The head of the screw is turned in such a way that its end touches the plane plate.

(4) Readings of the linear and circular scales are taken. Vernier or fractional part is found out by multiplying the circular scale reading with least count. Total reading is calculated by adding the linear scale reading and the fractional portion.

(5) Several readings are taken following the above procedures and mean value is found out.

(6) Now place the spherometer on the curved surface. The screw-head is raised slowly and is placed in such a way that its front-end touches the maximum/minimum point of the curved surface. Following the above procedure, the linear scale and circular scale readings are taken and the mean value is found out. Repeating the above procedure a few numbers of readings are taken and the average value is calculated.

The difference between these two readings gives the value h.

Fig: view of the lens and spherometer from the side

(7) Using the meter scale distances between the three legs are found out and the mean value is calculated. This gives the value of d.

(8) Now using the values of h and d in the above equation R is found out.

Determination of Least count

• Value of the minimum division of the linear scale = x mm
• Pitch = x mm
• Total number of divisions of the circular scale = y
• Least count, K = Pitch/Total no. of circular division = (x/y) mm = … mm
• Distance between the three legs of the spherometer = d mm.

Precautions

• The screw should move freely without friction.
• The screw should be moved in the same direction to avoid the back-lash error of the screw.
• Excess rotation should be avoided.

Sources of error

• The screw may have friction.
• The spherometer may have a back-lash error.
• Circular (disc) scale divisions may not be of equal size.

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Spherometer Experiment

Experiment: Determine the radius of curvature of a concave mirror using a spherometer.

Spherometer

When a spherometer is placed on a curved surface such that all its legs are touching it, the middle leg will be a little higher or lower than the plane of the outer legs by a small amount h which is related to R, the radius of curvature of the surface.

AH = a, the distance between the central leg and the outer leg.

From geometry,

AH × HB = GH × HE

a × a = h (2R – h)

a 2 = 2Rh – h 2

2RH = a 2 + h 2

R = a 2 /2h + h 2 /2h

R = a 2 /2h + h/2

H is the centre of equilateral triangle formed by the outer legs A, B, C.

cos 30° = AM/AH

√3/2 = l/2a

R = l 2 /6h + h/2

Material Required

Spherometer, plane glass slab, concave mirror, half metre rod

How To Perform the Experiment

1. Examine the spherometer, noting carefully that the legs and the vertical scale are not shaky and that the central screw is not very loose.

2. Find the pitch of the screw by determining the vertical distance covered in 4 or 5 rotations.

Pitch = Distance moved / No. of complete rotations

3. Find the least count by dividing the pitch by number of divisions on the circular scale.

Least count = Pitch of the screw / No. of divisions on circular scale

4.  Set the given concave mirror on a horizontal surface firmly and place the spherometer on it and adjust the central leg till it touches the surface. All the four legs touch the surface of the concave mirror.

5. In order to eliminate back-lash error, proceed slowly as the central leg reach close to the mirror surface. Stop when central leg touches the mirror surface and the entire spherometer just rotates, hanging on the central leg.

6. Read the coincident division on the circular scale and also the main scale reading on the vertical scale. Thus find the total reading.

7. Now place the instrument on the surface of the plane glass slab and find how many complete turns have to be made to bring the tip of the central leg to the plane of the outer leg. Also read the coincident division on the circular scale. Thus find the total reading on the glass slab. The difference between the above two readings gives h.

8. Press the spherometer gently on the notebook so as to get pricks of the feet which are pointed. Measure the distance between each pair of outer pricks and find their mean. This gives l.

Sources of Errors

• By spherometer we find R of front surface of the mirror. But its back surface is polished.
• Since l is very small, an error in it causes large percentage error in the result.
• Back-lash error is eliminated only by the weight of the spherometer. Since it is a small weight, back-lash error may be only partially eliminated by it.

Spherometer: Determining Radius of Curvature of Spherical surface

What is a spherometer? How do you use it to determine the radius of curvature of a given spherical surface?

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Target Exam ---

To determine the radius of curvature of a given spherical surface by a spherometer.

Apparatus Spherometer, convex surface (it may be an unpolished convex mirror), a big size plane glass slab, or plane mirror.

Theory It works on the principle of micrometer screw. It is used to measure either very small thickness or the radius of curvature of a spherical surface which is why it is called a spherometer.

• Raise the central screw of the spherometer and press the spherometer gently on the practical notebook so as to get pricks off the three legs. Mark these pricks as A, B, and C.
• Measure the distance between the pricks (points) by joining the points to form a triangle ABC.
• Note these distances (AB, BC, AC) in the notebook and take their mean.
• Find the value of one vertical {pitch) scale division.
• Determine the pitch and the least count of the spherometer [Art. 2.13(c)] and record it step-wise.
• Raise the screw sufficiently upwards.
• Place the spherometer on the convex surface so that its three legs rest on it.
• Gently, turn the screw downwards till the screw tip just touches the convex surface. (The tip of the screw will just touch its image on the convex glass surface).
• Note the reading of the circular (disc) scale which is in line with the vertical (pitch) scale. Let it be a (It will act as a reference).
• Remove the spherometer from over the convex surface and place it over a large size plane glass slab.
• Turn the screw downwards and count the number of complete rotations (n 1 ) made by the disc (one rotation becomes complete when the reference reading crosses past the pitch scale).
• Continue till the tip of the screw just touches the plane surface of the glass slab.
• Note the reading of the circular scale which is finally in line with the vertical (pitch) scale. Let it be b.
• Find the number of circular (disc) scale divisions in the last incomplete rotation.
• Repeat steps 6 to 14, three times. Record the observation in tabular form.

Result The radius of curvature of the given convex surface is cm.

Precautions

• The screw should move freely without friction.
• The screw should be moved in the same direction to avoid the back-lash error of the screw.
• Excess rotation should be avoided.

Sources of error

• The screw may have friction.
• The spherometer may have a back-lash error.
• Circular (disc) scale divisions may not be of equal size.

Question.1. Describe the principle of a spherometer. Answer. It works on the principle of a micrometer screw.

Question.2. Why is a spherometer so called? Answer. It measures the radius of curvature of spherical surfaces, hence it is called a spherometer.

Question.4. What are the values of P and R? for a plane surface? Answer. For a plane surface, P = 0 and R = infinite.

Question.5. What is meant by the pitch of the spherometer? Answer. The pitch is the distance between two consecutive threads of the screw taken parallel to the axis of rotation or the distance moved by the screw in one complete rotation of the circular scale.

Question.6. How can the accuracy of a spherometer be increased? Answer. The smaller is the least count, the more is the accuracy of an instrument and vice versa. The accuracy of the spherometer can be increased by decreasing the pitch or by increasing the number of divisions of the circular scale.

Question.7. The least count of screw gauge and spherometer is the same. Which will you prefer to measure the radius of curvature of the lens or mirror? Answer. The spherometer.

Question.9. Can you measure the focal length of a lens? Answer. Yes, by measuring R 1 and R 2 by spherometer n 1 = 1 and n 2 is known refractive index of the material of the lens.

Question.10. Why is the good spherometer made of gunmetal? Answer. To minimize wear and tear.

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NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

To Determine Radius of Curvature of a Given Spherical surface by a Spherometer

November 22, 2016 by Bhagya

Physics Lab Manual NCERT Solutions Class 11 Physics Sample Papers Aim To determine radius of curvature of a given spherical surface by a spherometer.

Apparatus  Spherometer, convex surface (it may be unpolished convex mirror), a big size plane glass slab or plane mirror.

Theory  It works on the principle of micrometre screw (Section 2.09) It is used to measure either very small thickness or the radius of curvature of a spherical surface that is why it is called a spherometer.

• Raise the central screw of the spherometer and press the spherometer gently on the practical note-book so as to get pricks of the three legs. Mark these pricks as A, B and C.
• Measure the distance between the pricks (points) by joining the points as to form a triangle ABC.
• Note these distances (AB, BC, AC) on notebook and take their mean.
• Find the value of one vertical {pitch) scale division.
• Determine the pitch and the least count of the spherometer [Art. 2.13(c)] and record it step wise.
• Raise the screw sufficiently upwards.
• Place the spherometer on the convex surface so that its three legs rest on it.
• Gently, turn the screw downwards till the screw tip just touches the convex surface. (The tip of the screw will just touch its image in the convex glass surface).
• Note the reading of the circular (disc) scale which is in line with the vertical (pitch) scale. Let it be a (It will act as reference).
• Remove the spherometer from over the convex surface and place over a large size plane glass slab.
• Turn the screw downwards and count the number of complete rotations (n 1 ) made by the disc (one rotation becomes complete when the reference reading crosses past the pitch scale).
• Continue till the tip of the screw just touches the plane surface of the glass slab.
• Note the reading of the circular scale which is finally in line with the vertical (pitch) scale. Let it be b.
• Find the number of circular (disc) scale division in last incomplete rotation.
• Repeat steps 6 to 14, three times. Record the observation in tabular form.

Result  The radius of curvature of the given convex surface is cm.

Precautions 

• The screw should move freely without friction.
• The screw should be moved in same direction to avoid back-lash error of the screw.
• Excess rotation should be avoided.

Sources of error 

• The screw may have friction.
• The spherometer may have back-lash error.
• Circular (disc) scale divisions may not be of equal size.

Question.1. Describe principle of a spherometer. Answer. It works on the principle of micrometre screw. ‘

Question.2. Why is a spherometer so called ? Answer. It measures radius of curvature of spherical surfaces, hence it is called a spherometer.

Question.4. What are values of P and R? for a plane surface ? Answer. For a plane surface, P = 0 and R = infinite.

Question.5. What is meant by pitch of spherometer ? Answer. The pitch is the distance between two consecutive threads of the screw taken parallel to the axis of rotation or the distance moved by the screw in one complete rotation of the circular scale.

Question.6. How can the accuracy of a spherometer be increased ? Answer. The smaller is the least count, the more is the accuracy of an instrument and vice versa. The accuracy of the spherometer can be increased by decreasing the pitch or by increasing the number of divisions of circular scale.

Question.7. The least count of screw gauge and spherometer is same. Which will you prefer to measure the radius of curvature of lens or mirror ? Answer. The spherometer.

Question.9. Can you measure the focal length of a lens ? Answer. Yes, by measuring R 1 and R 2 by spherometer n 1 = 1 and n 2 is known refractive index of material of lens.

Question.10. Why are the good spherometer made of gun metal ? Answer. To minimise wear and tear.

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Expert strategies that (really) help kids reduce screen time

Four strategies to reset your family’s relationship with screen time from the surgeon general, parents and researchers who know what kids actually do online

When researchers last year tracked the smartphones of 200 volunteers ages 11 to 17, they found teens weren’t just mindless screen zombies.

In fact, they used all sorts of strategies to try to disconnect: Some left on “do not disturb” to silence never-ending notifications. Others charged their phones outside their bedrooms to help them get sleep.

Turns out, teens want time away from phones and social media, too.

If it feels as though everyone in your family is spending too much time looking at screens instead of each other, you’re not alone. Giant corporations are working against all of us to make social media, games and apps ever harder to put down. U.S. Surgeon General Vivek Murthy wrote in op-ed in the New York Times Monday that the “mental health crisis among young people is an emergency — and social media has emerged as an important contributor.”

How to diagnose unhealthy tech use

“There is not a generational decline in willpower,” Murthy said in an interview earlier this year. “The platforms are designed specifically to maximize how much time we spend on them.”

The good news is that it’s never too late for a family tech reset. But you won’t find the solution only in parental controls and screen-time restrictions. The most effective approach is to listen to what your kids say about their online experiences and make rebalancing a project for the whole family. That includes parents taking a hard look at their own phone habits, too.

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Sick of scams? Stop answering your phone.

The four-step strategy we recommend below is based on interviews with doctors, parents and researchers who have studied what’s actually going on behind kids’ screens — and seen what really works to change behavior. This is targeted primarily at families with teens who already have phones or other devices. (When to give a kid their first phone or social media account is an important separate conversation.)

Parents have plenty of reason to be concerned about mental health, bullying and exploitation online. But try to remember: Screens aren’t always the enemy, even if children are using them differently from how you did growing up. Many teens find community online. For some, it’s a constructive and even lifesaving place to explore identity.

That is why the first step is to practice more empathy and less eye-rolling, said Emily Weinstein, executive director of Harvard’s Center for Digital Thriving .

1. Have an awkward conversation

Parents have never been teenagers in the age of TikTok and artificial intelligence. So before you launch into rulemaking, it’s essential to do some fact-finding.

Talk to your teen about their online life the same way you would talk about school or extracurriculars. Where do they like to spend time? What need does a particular app or game fulfill? What real-world activities do they value, and how can tech get in the way?

To get the conversation started, call a casual family meeting. Tell your teen you want to hear their thoughts about the family’s tech use — what’s going well, and where they could use your help. Make sure you’re open to feedback about your tech use, as well.

Listen to what your kids already do on their own to tame their screen time. A good question Weinstein learned from her research with teens: What do you do when you really want to focus on someone or something you care about?

1 in 10 teens already use ChatGPT for school. Here’s how to guide them.

Call out what researchers call “technoference”: the moments when technology interferes with relationships and actual human connection.

As you process this information together, focus on building your understanding rather than immediately giving advice. Were you bullied as a teen? Were you curious about sex and other topics that were “too mature” for you? Your child is not reinventing the wheel here.

While interviewing dozens of teenagers for their book “Behind Their Screens: What Teens Are Facing (and Adults Are Missing),” Weinstein and fellow researcher Carrie James were surprised to learn that many teenagers slept with their phones because they were worried about missing a text from a friend in crisis. If frustrated parents viewed the habit as simple “phone addiction,” they would be missing out on an important part of their child’s inner world, Weinstein said.

Let the family meeting be sacred ground where teens can share without getting in trouble.

2. Conduct some screen time ‘experiments’

Now you need an action plan. Critically, though, it shouldn’t just be rules handed down by parents. Think of it, instead, as experiments.

As a family, brainstorm some ways to reclaim your time and focus from devices. Failure is okay.

The focus should be as much about reducing screen time as it is replacing that time with something you would like more of, be it family adventures or sleep. “We can’t just expect that we’re going to reduce screen time by an hour and things are just going to get better in our family,” University of Michigan pediatrician Jenny Radesky says. “Then you just leave this vacuum of an hour where kids are going to be like, ‘I’m bored, I don’t know what to do.’”

Ideas from the experts include:

• No devices out at meal times, so everyone gets to look at one another.
• When you’re hanging out, have everyone put their phone in a stack in the middle of the table, with some silly consequence for the person who looks first.
• No devices out in the car, so you can have conversations instead.
• No devices in or by the bed at night, so it’s easier to sleep. Instead, charge them in a common space in the home.
• Set the house WiFi router to shut off data access at an agreed time each night.
• Try listening to an audiobook or podcast together.
• Pick a vacation destination that is fun for everyone, but agree in advance that you won’t carry your phones with you — or just pick a place that has little or no cell and WiFi service.

And before you start, pick a date on the calendar to agree to all get back together and talk about how the experiments affected everyone’s mood. Talk about how it felt without the “digital pacifier” of a screen to look at in moments of awkwardness or social uncertainty — and whether it ended up giving you moments to connect.

3. Agree on rules — that parents have to follow, too

When you find some experiments that work, turn them into rules that everyone agrees on.

Together, write a family tech plan ( like these samples from Common Sense Media ). Kids tend to know what’s right and wrong, and you can help them fill in the blanks.

Then you have to follow the rules, too. “Parents are the number one role models for their kids when it comes to technology,” says Jim Steyer , founder of Common Sense Media.

If you’re not present and engaged, why should your kid be? They see you when you’re answering work Slacks while they are telling you about their day, or texting while you’re driving. In 2023, American adults used their phones an average of 4.3 hours per day, according to research firm Data.ai.

If you do need to pick up your phone during a communal moment for important grown-up reasons, say out loud why you’re doing so — like, “I’m looking up directions right now.”

4. Review safety tools together

Most social media and gaming sites come with safety and privacy settings. And while they won’t fix systemic issues with abuse and social media amplifying harmful information, they’re worth a regular checkup.

Where possible:

• Turn off the ability for people outside your teen’s friend circle to direct message, mention or tag them.
• Turn profiles to private so your teen has to accept new friends before they can engage.
• On TikTok and Instagram, turn off “stitch” or “remix” so strangers can’t boost their videos to new audiences.

Admittedly, it’s tough to keep tabs on your teen’s social media experience from the outside. Dozens of parental-monitoring tools promise to scan your child’s messages for “inappropriate” content or feed you updates on their activity. Some parents follow their kids online from secret accounts — or just poke around teens’ phones when they’re not looking.

Our experts agree that surveilling a teen who isn’t already in deep crisis can do more harm than good. Even if you don’t trust them, it’s important they trust you, Radesky says. Instead of spying, tell your teen exactly what you will do to stay apprised of their online life and where you will grant them privacy.

Most importantly, help teens identify what behavior is acceptable. Brainstorm together what type of online flirtation is appropriate. Discuss what they would do if someone made them feel uncomfortable, guilty, scared or attacked. And make yourself available to gut-check their posts before they hit send.

The body can be a useful tool here. Help kids learn to take signals from their bodies by modeling it yourself. “Wow, time on that app really made me feel energized,” or “Gosh, my chest feels really anxious after watching that video.”

If you’re tired of constantly playing “bad cop” with your teen’s tech use, we’ve got some good news: Striking the right balance is challenging for you both, and you can navigate this new world as a team.

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