Question

The upper end of a wire of radius $4mm$ and length $100cm$ is clamped and its other end is twisted through an angle of $30 rad $. The angle of shear is:
A) $12 rad $
B) $1.2 rad $
C) $0.12 rad $
D) $0.012 rad $

Answer 1

Hint: Use the relation between angle of shear and angle of twist. We are given all the required values, after putting them in the formula, we just need to simplify and perform basic calculation and we will get the answer.

Formula Used:
The relation between angle of shear, $\phi $ and angle of twist, $\theta $ is given by the equation $r\theta = l\phi $ where, $r$ is the radius of wire and $l$ is the length of the wire.

Complete step by step solution:
Let us take a look at all the given values. Radius of the wire $r = 4mm$ , length of the wire $l = 100cm$ , angle of twist $\theta = 30 rad $
Before solving the question, we need to convert all these values into their respective SI units.
Now we know that the SI unit of length is metre. Hence, we will convert the radius and length of the wire into metres. We get, $r = 0.004m$ and $l = 0.1m$

We know that the relation between angle of shear, $\phi $ and angle of twist, $\theta $ is given by the equation $r\theta = l\phi $ where, $r$ is the radius of wire and $l$ is the length of the wire.
On putting values in the formula, we get $0.004 \times 30 = 0.1 \times \phi $
Which implies, $\phi = \dfrac{{0.004 \times 30}}{{0.1}}$ or, $\phi = \dfrac{{4 \times {{10}^{ - 3}} \times 30}}{{1 \times {{10}^{ - 1}}}}$
Which gives, $\phi = \dfrac{{4 \times 30}}{{{{10}^{2}}}}$
This implies, $\phi = \dfrac{{120}}{{{{10}^{2}}}} = 1.2 rad $

Hence, option B is the correct answer.

Note: In questions like these, make sure all the values are given in SI units. If not, convert the given values into SI units before using them in the formula. If we somehow do not convert the values in Si units, we will get a different answer. In this case, had we forgotten to convert the values into SI units, then the answer would have been different in only the decimal places and we would have seen the option in the answer and written that as the correct option. Always convert values into SI units before using them in a question.