Question

A wire of length 5 m is twisted through${30^0}$ at the free end. If the radius of wire is 1 mm, the shearing strain in the wire is:
A) ${30^0}$
B) 0.36’
C) ${1^0}$
D) ${0.18^0}$

Answer 1

Hint:In this question we are already given that a wire of length 5 m is twisted through 30 degrees. We will simply find out the shearing stress along 5 m and then convert it to millimeter and then eventually find out the same shearing stress of wire along 1 mm. lastly simplifying it by multiplying by 60 to convert it into seconds.

Complete solution:
Shear stress: - Shear stress, force tending to cause deformation of a material by slippage along a plane or planes parallel to the imposed stress. The resultant shear is of great importance in nature, being intimately related to the down slope movement of earth materials and to earthquakes.
Here, we are given length l=5m
The wire is twisted through ${30^0}$ at the free end, this implies $\phi = {30^0}$
As the wire is twisted then we will convert it to the radian which is 1 rad=${\left( {\dfrac{{180}}{\pi }} \right)^0}$
Then, 30×$\dfrac{\pi }{{180}}$, this is equals to $\dfrac{\pi }{6}rad$
Now, we need to find the shearing strain when radius is 1 mm
Then, $5\left( \theta \right)$=${10^{ - 3}}\dfrac{\pi }{6}$ so $\theta $=${10^{ - 3}}\dfrac{\pi }{{30}}$
On further solving the above equation and opening the value of pi we get, $\left( {1.0466} \right) \times {10^{ - 4}} \times \dfrac{{180}}{\pi }$
Converting it into second and multiplying by 60=$59.9 \times {10^{ - 4}} \times 60\min $
This is equals to 0.359sec or $ \approx $ 0.36’
Hence, the radius of wire is 1 mm, the shearing strain in the wire is: 0.36’

Option B is the correct answer.

Additional information:Shear strain is measured as a change in angle between lines that were originally perpendicular. Consider an elemental area that undergoes a distortion that produces angular changes, but which leaves the sides of the area approximately the same length
Examples of shear stress:
-While Chewing food between the teeth.
-While walking or running while our feet push the ground back to move forward.
-When a moving vehicle starts or stops, the surface of the seat experiences shear stress.
-When water flows River beds experience shear stress.

Note:Degrees (a right angle is 90 degrees) and radian measure (a right angle is 100 grads) have their uses. The length of the arc subtended by the central angle becomes the radian measure of the angle. This keeps all the important numbers like the sine and cosine of the central angle, on the same scale.