Question

A rod of length \[l\] and radius \[r\] is joined to a rod of length \[\dfrac{l}{2}\] and radius \[\dfrac{r}{2}\] of same material. The free end of small rod is fixed to a rigid base and the free end of larger rod is given a twist of \[{\theta ^ \circ }\] ​, the twist angle at the joint will be
A. $\dfrac{\theta }{4}$
B. $\dfrac{\theta }{2}$
C. $\dfrac{{5\theta }}{6}$
D. $\dfrac{{8\theta }}{9}$

Answer 1

Hint: Before we go into the question, it's important to understand what an angle of twist is. The angle of twist of a shaft under torsional loading is the angle through which the fixed end rotates with respect to the free end.

Complete answer:
In order to answer the question, let us first write all the given values accordingly;
Length of the rod which is joined is \[l\]
Radius of the rod which is joined is \[r\]
Now, length of the rod to which another rod is joined is \[(l') = \dfrac{l}{2}\]
And, radius of that rod to which another rod joined is \[\left( {r'} \right) = \dfrac{r}{2}\]
Now, we need to calculate the twist angle at the joint as per our question.
Hence, using the formula of torque
$\tau = c\theta $
Plug in the value in the formula.
\[\tau = \dfrac{{\pi \eta {r^4}\theta }}{{2l}} = constant\]
Therefore, for both the rod
$\dfrac{{\pi \eta {r^4}\left( {\theta - {\theta _0}} \right)}}{{2l}} = \dfrac{{\pi \eta {{\left( {\dfrac{r}{2}} \right)}^4}\left( {{\theta _0} - \theta '} \right)}}{{2\left( {\dfrac{l}{2}} \right)}}$
Evaluating the equation $\pi \eta $ will cancel out each other on both the sides
$ \Rightarrow \dfrac{{\theta - {\theta _0}}}{{2l}} = \dfrac{{{\theta _0}}}{{16l}}$
Now, \[l\] will be cancelled out on both sides.
$\Rightarrow \dfrac{{\theta - {\theta _0}}}{2} = \dfrac{{{\theta _0}}}{{16}}$
$\Rightarrow {\theta _0} = \dfrac{8}{9}\theta $
Therefore, the twist angle at the joint will be $\dfrac{{8\theta }}{9}$

So, the correct option is: (D) $\dfrac{{8\theta }}{9}$

Note:
The terms angle of twist and angle of shear should not be confused by students. When an item is subjected to shearing stress or deformation force, the angle of shear is defined as the angle of deformation that occurs on the sides. The angle of twist is the angle at which a rotating machine element spins or twists in relation to its free end.