Answer to Question #100966 in Physics for Emmanuel

Question

8(a). Explain the principle of the vernier calipers
(b). The figure below shows a vernier calliper which was used to measure the
diameter of a cylinder. What is the diameter d of the cylinder?







(c). What is the error in d.
(d). Deduce the percentage error in d
(e). Calculate the percentage error in the cross sectional area A
(f). Why is it better to take several readings for the diameter than to depend on a
single reading?

Answer 1

a) The vernier caliper implies using two scales to measure the size:

Above is the main scale with numbers, below is the vernier scale without numbers.

After you put the object inside the jaws, determine the distance between 0 on the main scale (above) and the first line on the vernier (yellow). It is 0.7 cm, or 7 mm.

Then find where the two lines from both scales coincide with each other (blue). Count from left to right along the vernier starting with 0. It is 0, 1, 2, 3.

b) That is why the reading (in centimeters in this example) is

$$d=0.7+3/10=0.73\text{ cm}.$$

c) A typical error is a half of the smallest count, i.e. $\Delta d=0.1/2=0.05 \text{ mm}.$

d) The percentage error:

$$\epsilon=\frac{\Delta d}{d}\cdot100\%=\frac{0.05}{7.3}\cdot100\%=1\%.$$

e) The area:

$$A=\pi d^2/4=41.85\text{ mm}^2.$$

The percentage error in area:

$$\frac{\Delta A}{A}=2\cdot\frac{\Delta d}{d}=2\epsilon=2\%.$$

f) Because one reading can be wrong. However, if you make 10 measurements in total, that one wrong measurement can be lost among other nine more or less correct.