# Find the volume of the Sphere – Vernier Calipers.

2 Q : Find the volume of the given sphere using vernier calipers.

Ans:

Formula :

1. Volume of the Sphere V =  $\frac{4}{3} \pi r^3 cm^3$,

V= volume of Sphere, r = radius of  Sphere
.

2.Least count of vernier calipers L.C = $\frac{S}{N}$ cm,

S = value of 1 Main scale division , N = Number of vernier divisions.

3.Length (or) diameter  of Cylinder = Main scale reading (a) cm + ( n*L.C ) cm.

n = vernier coincidence .

Procedure : First we have to determine the least count count of the given vernier calipers.

To determine the volume of the  Sphere we have to determine the radius  (r) of the cylinder and substituting this value in the equation for the volume of the Sphere we can calculate it.

a) To determine the diameter of the Sphere : Given Sphere is held gently between jaws 1,1 of the vernier calipers.The reading on the main scale just before the zero of the vernier is noted.This is called Main scale reading (M.S.R).The number of division (n) on the vernier which coincides perfectly with any one of the main scale divisions is noted.This is called vernier coincidence (V.C).The vernier coincidence (V.C=n) is multiplied by least count to get the fraction of a main scale division.This is added to the main scale reading (M.S.R) to total reading or total diameter of the sphere.

Total reading = M.S.R + ($V.C\times L.C$)

Take the readings,keeping the Sphere between jaws 1,1 at different positions.Post the values of M.S.R and vernier coincidence (n) in the table.Take at least 5 readings, get the average of these 5 readings which is mean diameter (d)of the Sphere.

Place the Sphere diametrically between the jaws 1,1 of the vernier calipers, post the values of M.S.R and vernier coincidence (n) in the table. Take at least 5 readings, calculate the average of these readings which gives the mean diameter ( d=2r ) of the Sphere.

c) To determine the volume of the Sphere :Substituting the value mean radius ( r) of the sphere which is already determined, in the formula V = $\frac{4}{3} \pi r^3 cm^3$,

Determine Least count of vernier calipers : From the given vernier calipers

S= Length of Main scale division = 1 mm = 0.1 cm,

N = Number of vernier scale divisions = 10 ,

Substitute these values in the formula of Least count L.C = $\frac{S}{N}$ = $\frac{0.1}{10}$ =0.01 cm.

Table for  Diameter of the Sphere :

 S.No M.S.R                acm Vernier Coincidence   (n) Fraction   b=n*L.C Total Reading (a+b) cm 1. 1.9 7 0.05 1.97 2. 1.9 6 0.04 1.96 3. 1.9 6 0.06 1.96 4. 1.9 7 0.05 1.97 5. 1.9 7 0.06 1.97

Average diameter of the sphere  d = 2r =  $\frac{(1.97+1.96+1.96+1.97+1.97)}{5}$ cm = $\frac{9.83}{5}$

Average radius of the sphere r =$\frac{d}{2}$ = $\frac{1.966}{2}$cm = 0.98 cm.

Observations :

Average radius of the cylinder r = 0.98 cm.

Calculations : Volume of the sphere V = $\frac{4}{3} \pi r^3 cm^3$ = $\frac{4}{3}\times\frac{22}{7}\times(0.98)^3$ $cm^3$

=3.94 $cm^3$

Precautions : 1) Take the M.S.R  and vernier coincide every time without parallax error.

2)Record all the reading in same system preferably in C.G.S system.

3) Do not apply excess pressure on the body held between the jaws.

4) Check for the ZERO error.When the two jaws of the vernier are in contact,if the zero division of the main scale coincides with the zero of the vernier scale no ZERO error will be there.If not ZERO error will be there, apply correction.

Result and Units : Volume of the sphere V = 3.94 $cm^3$.

# Find the volume of cylinder – Vernier calipers.

1 Q : Find the volume of the  given cylinder  using vernier calipers.

Ans :

Formula :

1. Volume of the cylinder V = $\pi r^2 l$ $cm^3$,

V= volume of cylinder, r = radius of cylinder  l = length of cylinder.

2.Least count of vernier calipers L.C = $\frac{S}{N}$ cm,

S = value of 1 Main scale division , N = Number of vernier divisions.

3.Length (or) diameter  of Cylinder = Main scale reading (a) cm + ( n*L.C ) cm.

n = vernier coincidence .

Procedure : First we have to determine the least count count of the given vernier calipers.

To determine the volume of the  cylinder we have to determine a)the length of the cylinder and b) radius of the cylinder and substituting these values in the equation for the volume of the cylinder we can calculate it.

a) To determine the length of the cylinder : Given cylinder is held gently between jaws 1,1 of the vernier calipers.The reading on the main scale just before the zero of the vernier is noted.This is called Main scale reading (M.S.R).The number of division (n) on the vernier which coincides perfectly with any one of the main scale divisions is noted.This is called vernier coincidence (V.C).The vernier coincidence (V.C=n) is multiplied by least count to get the fraction of a main scale division.This is added to the main scale reading (M.S.R) to total reading or total length of the cylinder.

Total reading = M.S.R + ($V.C\times L.C$)

Take the readings,keeping the cylinders between jaws 1,1 at different positions.Post the values of M.S.R and vernier coincidence (n) in the table.Take at least 5 readings, get the average of these 5 readings which is mean length(l )of the cylinder.

b)To determine the diameter of the cylinder : Place the cylinder diametrically between the jaws 1,1 of the vernier calipers, as in the above case post the values of M.S.R and vernier coincidence (n) in the table. Take at least 5 readings, calculate the average of these readings which gives the mean diameter ( d=2r ) of the cylinder.

c) To determine the volume of the cylinder :Substituting the values of mean length (l ) of the cylinder and mean diameter ( r) of the cylinder which is already determined, in the formula V = $\pi r^2 l cm^3$.

Determine Least count of vernier calipers : From the given vernier calipers

S= Length of Main scale division = 1 mm = 0.1 cm,

N = Number of vernier scale divisions = 10 ,

Substitute these values in the formula of Least count L.C = $\frac{S}{N}$ = $\frac{0.1}{10}$ =0.01 cm.

Table – Length of the cylinder :

 S.No M.S.R                acm Vernier Coincidence   (n) Fraction   b=n*L.C Total Reading (a+b) cm 1. 2.6 9 0.09 2.69 2. 2.7 1 0.01 2.71 3. 2.7 2 0.02 2.72 4. 2.7 2 0.02 2.72 5. 2.6 8 0.08 2.68

Average length of the cylinder l = $\frac{2.69+2.71+2.72+2.72+2.68}{5}$ cm=$\frac{13.52}{5}$ = 2.70 cm

b) Diameter of the cylinder :

 S.No M.S.R                acm Vernier Coincidence   (n) Fraction   b=n*L.C Total Reading (a+b) cm 1. 1.4 5 0.05 1.45 2. 1.4 4 0.04 1.44 3. 1.4 6 0.06 1.46 4. 1.4 5 0.05 1.45 5. 1.4 6 0.06 1.46

Average diameter of the cylinder  d = 2r = $\frac{1.45+1.44+1.46+1.45+1.46}{5}$ cm=$\frac{7.26}{5}$ = 1.45 cm,

Average radius of the cylinder r =$\frac{d}{2}$ =$\frac{1.45}{2}$ cm = 0.73 cm.

Observations :

Average length of the cylinder l = 2.70 cm,

Average radius of the cylinder r = 0.73 cm.

Calculations : Volume of the cylinder V = $\pi r^2 l$ $cm^3$ = $\frac{22}{7}\times(0.73)^2\times2.70$ $cm^3$ = 4.52 $cm^3$

Precautions : 1) Take the M.S.R  and vernier coincide every time without parallax error.

2)Record all the reading in same system preferably in C.G.S system.

3) Do not apply excess pressure on the body held between the jaws.

4) Check for the ZERO error.When the two jaws of the vernier are in contact,if the zero division of the main scale coincides with the zero of the vernier scale no ZERO error will be there.If not ZERO error will be there, apply correction.

Result and Units : Volume of the cylinder V = 4.52 $cm^3$.

# Vernier Calipers

Aim: To determine i) The volume of the given cylinder by measuring its length and diameter

ii ) The volume of the given sphere by measuring its diameter.

Apparatus : Vernier Calipers,Cylinder and sphere.

Description of Vernier Calipers: A Vernier calipers consists of mainly two parts i) A 2cm wide 15cm long rectangular metal strip .The left end bottom side of this strip consists of a fixed jaw 1 (A) and at the same end jaw 2(C) at the top of this strip. On the strip a scale (5)is graduated in Inches along the upper edge and another scale(4) is graduated in Centimeters along the lower edge. This is called Main Scale ‘S’ .

ii) A metal frame V called vernier slides over the Main Scale ‘S’ . At the bottom of this frame V a button 8(P) is attached,which helps to fix this vernier at any desired place on the main scale.This verier frame consists of jaw1 (B) at the bottom and a jaw 2(D) at the top .Two scales are graduated on this frame corresponding to two scales on the Main Scale ‘S’. The two scales 6 and 7 on the vernier are called Vernier scale.Vernier scale consists of equal number of divisions. When we move vernier frame over the main scale, a thin strip (3) will be projected out.The projection will be exactly equal to the distance between Jaws 1(AB) i.e the thickness of the object between jaws.

The lower jaws 1,1(AB) are used to measure the thickness or external diameter of the tubes,cylinders or spheres.

The upper jaws 2,2 (CD) are used to measure the inner diameters of hallow bodies like tubes or holes.

The thin strip ( 3) is used to measure the depth of the objects like test tubes.

Theory :  Principle of vernier calipers – N divisions on the vernier scale is equal to (N-1) divisions on the main scale.

N V.S.D = (N-1) M.S.D

1 V.S.D = $\frac{(N-1)}{N}$ M.S.D

Least count (L.C) of vernier calipers : Minimum length or thickness measurable with the vernier calipers is called its least count.

Least count (L.C) = 1 M.S.D – 1 V.S.D

L.C = 1 M.S.D – $\frac{(N-1)}{N}$ M.S.D

L.C = 1 M.S.D [ 1-$\frac{(N-1)}{N}$]

L.C = $\frac{1 M.S.D}{N}$  = $\frac{S}{N}$

Where S is the value of  one Main scale division and N is the number of equal divisions on the vernier scale.

Procedure : First we have to determine the least count count of the given vernier calipers.

To determine the volume of the  cylinder we have to determine a)the length of the cylinder and b) radius of the cylinder and substituting these values in the equation for the volume of the cylinder we can calculate it.

a) To determine the length of the cylinder : Given cylinder is held gently between jaws 1,1 of the vernier calipers.The reading on the main scale just before the zero of the vernier is noted.This is called Main scale reading (M.S.R).The number of division (n) on the vernier which coincides perfectly with any one of the main scale divisions is noted.This is called vernier coincidence (V.C).The vernier coincidence (V.C=n) is multiplied by least count to get the fraction of a main scale division.This is added to the main scale reading (M.S.R) to total reading or total length of the cylinder.

Total reading = M.S.R + ($V.C\times L.C$)

Take the readings,keeping the cylinders between jaws 1,1 at different positions.Post the values of M.S.R and vernier coincidence (n) in the table.Take at least 5 readings, get the average of these 5 readings which is mean length(l )of the cylinder.

b)To determine the diameter of the cylinder : Place the cylinder diametrically between the jaws 1,1 of the vernier calipers, as in the above case post the values of M.S.R and vernier coincidence (n) in the table. Take at least 5 readings, calculate the average of these readings which gives the mean diameter ( d=2r ) of the cylinder.

c) To determine the volume of the cylinder :Substituting the values of mean length (l ) of the cylinder and mean diameter ( r) of the cylinder which is already determined, in the formula V = $\pi r^2 l cm^3$.

d)To determine the diameter of the sphere : The given sphere is held firmly between jaws of the vernier calipers, in such a way the points where the jaws are in contact with sphere should be the two extremes of the chord of the cylinder.Post the values of the M.S.R and vernier coincidence (n) in the table . Take at least 5 readings, calculate the average of these readings which gives the mean diameter (d=2 r ) of the sphere.

e)To determine the volume of the sphere :Calculate the radius of the sphere  r = d/2 .Substitute the value of  mean radius (r)  in the formula of the volume of the sphere V = $\frac{4}{3} \pi r^3 cm^3$ .

Precautions to be taken while doing the experiment : 1) Take the M.S.R  and vernier coincide every time without parallax error. 2)Record all the reading in same system preferably in C.G.S system. 3) Do not apply excess pressure on the body held between the jaws. 4) Check for the ZERO error.When the two jaws of the vernier are in contact,if the zero division of the main scale coincides with the zero of the vernier scale no ZERO error will be there.If not ZERO error will be there, apply correction.

Observations:

I) Least count of Vernier calipers :

a) Value of 1 Main scale division  = 1 M.S.D = S = ……..  cm,

b) Number of divisions on the vernier scale   N= ……… cm,

Least count              L.C = $\frac{S}{N}$   = ………. cm.

c) Zero error =x ( positive error)

II) Volume of the Cylinder :

a)Length of the cylinder :

 S.No M.S.R                acm Vernier Coincidence   (n) Fraction   b=(n-x)*L.C Total Reading (a+b) cm 1. 2. 3. 4. 5.

Average length of the cylinder l = ……… cm.

b) Diameter of the cylinder :

 S.No M.S.R                acm Vernier Coincidence   (n) Fraction   b=(n-x)*L.C Total Reading (a+b) cm 1. 2. 3. 4. 5.

Average diameter of the cylinder  d = 2r = …………. cm,

Mean radius of the cylinder           r = d/2 = ………….. cm,

Volume of the cylinder                   V = $\pi r^2 l cm^3$.

III ) Volume of the sphere :

a) Diameter of the sphere :

 S.No M.S.R                acm Vernier Coincidence   (n) Fraction   b=(n-x)*L.C Total Reading (a+b) cm 1. 2. 3. 4. 5.

Average diameter of the sphere  d = 2r = …………. cm,

Mean radius of the sphere           r = d/2 = ………….. cm,

Volume of the sphere                   V = $\frac{4}{3} \pi r^3 cm^3$.

Result : 1. Volume of the given cylinder  V= ……….. $cm^3$

2.Volume of the given sphere      V= ……….. $cm^3$.

Watch this vedio :

Review Questions – Vernier calipers.

During practical examinations  the examiner can shoot any question related to the experiment you are doing.Here I will try to answer few Frequently asked questions by   examiner.

1Q: What is the advantage of vernier calipers over a regular  scale graduated in millimeters?

Ans: We can measure up to a millimeter with regular scale which is graduated in millimeters, Where as  with a vernier calipers with 10 equal divisions on the vernier scale we can measure up to 1/10 (0.1) mm accurately.

2Q:What is the principle of vernier ?

Ans: (N-1) main scale divisions  = N divisions on vernier scale,

(N-1) M.S.D = N V.S.D   is the principle of vernier.

3Q:What do you mean by least count of vernier calipers?

Ans: The difference  of 1 M.S.D  and 1 V.S.D is called least count of a vernier calipers. (or) The minimum

length which can be measured by a vernier calipers is called its least count.

4Q:What is the use of lower jaws of a vernier calipers?

Ans: The lower jaws or used to measure the  thickness of object,outer diameters of tubes,spheres   and  cylinders.

5Q:What is the use of upper jaws?

Ans: The upper jaws of the vernier calipers are used to measure the inner diameters of rings,tubes and inner diameters of hallow cyliders , hallow spheres.

6Q:What is the use of the thin strip moving behind the main scale?

Ans: The strip is used to measure  the depths of the tubes and level of the liquid inside tubes or jars.

7Q:Two different vernier calipers  have different number of equal divisions on their vernier scales i) 10 equal divisions ii) 50 equal divisions . Which can measure more accurately.

Ans: The least count of the I vernier calipers  = S/N = $\frac{1mm}{10}$ =0.1mm

The least count of the II vernier calipers  = S/N = $\frac{1mm}{50}$ =0.02mm.

That is the II vernier calipers with 0.02mm of L.C can measure up to 0.02mm accurately.Hence the accuracy of II vernier calipers in more than I vernier calipers.

8Q:What is the ZERO error of a vernier calipers?

Ans:When two lower jaws of the vernier calipers are in contact, if the zero division of main scale does not coincide with the zeroth division of vernier scale, such vernier calipers will have zero error.

9Q: What is the formula of least count?

Ans: Least count L.C = $\frac{S}{N}$.

10Q:What is the formula to calculate the volume of the cylinder?

Ans: Volume of the cylinder V = $\pi r^2 l$. , where r = radius of the cylinder and l = length  of the cylinder.

11Q:What is the formula to calculate the volume of a Sphere?

Ans: Volume of the Sphere V = $\frac{4}{3} \pi r^3$, where r = radius of  the sphere.

12Q:What is the formula to calculate total reading with a vernier calipers?

Ans: Total reading = M.S.R + $n\times L.C$.

13.How many types of Zero errors will be possible in vernier calipers?

Ans: Two types of Zero errors are possible in vernier calipers they are i) Negative Zero error ii) Positive Zero error

14. What is Positive zero error,how do we correct such an error in a vernier calipers?

Ans:When two lower jaws of the vernier calipers are in contact, if the zero division of main scale does not coincide with the zeroth division of vernier scale and the zero of the vernier is to the right of zero of main scale such an error is called Positive Zero error.So,the zero correction should be subtracted from the reading which is measured.

Ex:When two lower jaws of the vernier calipers are in contact, if the zero division of the vernier is to the right of  zeroth division of main scale and if the vernier coincidence is 1.Then the correction is subtraction i.e Total Reading – $1 \times L.C$

15.What is Negative zero error,how do we correct such an error in a vernier calipers?

Ans:When two lower jaws of the vernier calipers are in contact, if the zero division of main scale does not coincide with the zeroth division of vernier scale and the zero of the vernier is to the left of zero of main scale such an error is called negative Zero error.So,the zero correction should be added to the reading which is measured.

Ex:When two lower jaws of the vernier calipers are in contact, if the zero division of the vernier is to the left of  zeroth division of main scale and if the vernier coincidence is 8.Then the correction is addition i.e Total Reading + $8 \times L.C$