Question

The upper end of a wire $1\,m$ long and $2mm$ radius is clamped. The lower end is twisted through angle $45^\circ $. What is the angle of twist?
A. $0.009^\circ $
B. $0.09^\circ $
C. $0.9^\circ $
D. $9^\circ $

Answer 1

Hint: The torsion equation is used to compute the twist angle. A term used in the torsion equation to calculate the shaft length, rigidity modulus, and polar moment of inertia. The relationship between the angle of shear and the angle of twist must be used to calculate the angle of twist. We already have all the necessary data; after entering them into the formula, all we need to do is simplify and carry out some simple calculations to obtain the result.

Formula Used:
The relation between angle of shear, $\varphi $ 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:
In the question, we have given the radius of the wire $r = 2mm$ , length of the wire $l = 1m$ and angle of twist $\theta = 45^\circ $. Before solving the question, we need to convert all these values into their respective SI units. As we know that the SI unit of radius is metre. Hence, we will convert the radius into metres. We get, $r = 0.002m$

And, the relation between angle of shear, $\varphi $ and angle of twist, $\theta $ is given by the equation
$r\theta = l\phi $
Since the radius is very small. So, in this case on putting values in the formula, we get;
$0.002 \times 45 = 1 \times \phi $
Then, we have
$\phi = \dfrac{{0.002 \times 45}}{1}$
$\Rightarrow \phi = \dfrac{{2 \times {{10}^{ - 3}} \times 45}}{1}$
$\Rightarrow \phi = 0.002 \times 45$
$\therefore \phi = 0.09^\circ $

Therefore, the correct option is B.

Note: Make sure that all values are provided in SI units when answering questions like this. If not, then before using the specified values in the formula, convert them into SI units. If for any reason we do not convert the values into SI units, the result will be different. In this scenario, the response would have been different in just the decimal places if we hadn't remembered to convert the data into SI units. As a result, we would have seen the option in the answer and marked it as the correct one. Before utilising a value in a question, always convert it to SI units.