Question

A 2m long rod of radius 1 cm which is fixed from one end is given a twist of 0.8 radians the shear strain developed will be:
(a) 0.002
(b) 0.004
(c) 0.008
(d) 0.016

Answer 1

Hint: Strain is a measure of how much an object has deformed on the application of a force. More the deformation, more will be the strain experienced by the object. Shear strain is a type of strain. Shearing strain is due to a force applied tangentially upon the surface of the object. It is the ratio of deformation of a layer divided by its distance from the fixed layer(the other end of the rod in this case).


Formula Used:
1. Shear strain: \[\phi = \dfrac{{L'}}{L}\] ……(1)
Where,
L’ is the deformation of the wire after a force is applied on it
L is the original length of the wire when the force was not there
2. Deformation: $L' = r\theta $ ……(2)
Where,
r is that radius of wire
$\theta $ is the angle by which the wire is twisted


Complete step by step answer:
Given:
1. Length of wire L=2m
2. Radius of wire r=1cm
3. Angle of twist $\theta $=0.8 radians

To find: The shear strain developed in the wire.

Step 1:
Convert radius r to SI units:
$
  r = 1cm \times \dfrac{1}{{100}}m \\
   \Rightarrow r = 0.01m \\
 $

Step 2:
Calculate the deformation using eq (2):
$
   \Rightarrow L' = 0.01 \times 0.8 \\
   \Rightarrow L' = 0.008m.radian \\
 $

Step 3:
Calculate the shear strain using eq (3):
\[
  \phi = \dfrac{{0.008}}{2} \\
   \Rightarrow \phi = 0.004radian \\
 \]

Final Answer
The shear strain developed will be (b) 0.004.

Note: There are different types of strain, for example : Longitudinal Strain, Shearing strain and volume strain. Here we are dealing with shearing strain. Shearing strain is defined as the displacement of the layer divided by the distance from the fixed layer. Strain is a dimensionless quantity as it is the ratio of deformation and original length, both with the same units. In case of shear strain, we measure it in terms of radians, which is again a unitless quantity.