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A long rod of radius 1 cm and length 2 m which is fixed at one end is given a twist of 0.8 radian. The shear strain developed will be:
A) 0.001 radians
B) 0.004 radians
C) 0.002 radians
D) 0.04 radians.

Last updated date: 13th Jun 2024
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Hint:The shear strain is the ratio of change in deformation to the original length perpendicular to the shear stress. The shear strain occurs when the applied force is parallel or tangent to the surface area of the body. Torsional may be the cause of shear strain and also shear stress. Shear stress is the ratio of the force parallel to the surface area to the area of cross section.

Formula used:The formula of the shear strain is given by, $\phi = \dfrac{{r \cdot \theta }}{l}$ where $\phi $is shear strain r is the radius of the rod, l is the length of the radius and $\theta $ is the angle of twist.

Step by step solution:
It is given that the value of twist is equal to 0.8 radians, the radius is given as 1 cm and the length of the rod is 2 m.
Apply the formula for the shear strain for the rod,
$ \Rightarrow \phi = \dfrac{{r \cdot \theta }}{l}$
Where $\phi $is shear strain r is the radius of the rod and l is the length of the radius.
As the value of radius is $r = 1cm$, length $l = 2m$ and angle of twist is equal to $\theta = 0 \cdot 8{\text{ radian}}$.
$ \Rightarrow \phi = \dfrac{{r \cdot \theta }}{l}$
$ \Rightarrow \phi = \left( {\dfrac{1}{{100}}} \right) \cdot \left( {\dfrac{{0 \cdot 8}}{2}} \right)$
Converting radius from centimeter into meter as $1m = 100cm$.
$ \Rightarrow \phi = \left( {\dfrac{1}{{100}}} \right) \cdot \left( {\dfrac{{0 \cdot 8}}{2}} \right)$
$ \Rightarrow \phi = 0 \cdot 004{\text{ radians}}$
The shear strain is given by$\phi = 0 \cdot 004{\text{ radians}}$.

The correct answer for this problem is option B.

Note: The units of all the quantities should be the same while calculating the value of the shear strain otherwise the answer may come wrong. The unit of shear strain is always represented in radians, if the angle of twist is given in other units than radians then it is advisable to convert it into radians.