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Question: A rod of length \[l\] and radius \[r\] is joined to a rod of length \[\dfrac{l}{2}\] and radius \[\d...

A rod of length ll and radius rr is joined to a rod of length l2\dfrac{l}{2} and radius r2\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 θ∘{\theta ^ \circ } ​, the twist angle at the joint will be
A. θ4\dfrac{\theta }{4}
B. θ2\dfrac{\theta }{2}
C. 5θ6\dfrac{{5\theta }}{6}
D. 8θ9\dfrac{{8\theta }}{9}

Explanation

Solution

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 ll
Radius of the rod which is joined is rr
Now, length of the rod to which another rod is joined is (l′)=l2(l') = \dfrac{l}{2}
And, radius of that rod to which another rod joined is (r′)=r2\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
τ=cθ\tau = c\theta
Plug in the value in the formula.
τ=πηr4θ2l=constant\tau = \dfrac{{\pi \eta {r^4}\theta }}{{2l}} = constant
Therefore, for both the rod
πηr4(θ−θ0)2l=πη(r2)4(θ0−θ′)2(l2)\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 πη\pi \eta will cancel out each other on both the sides
⇒θ−θ02l=θ016l\Rightarrow \dfrac{{\theta - {\theta _0}}}{{2l}} = \dfrac{{{\theta _0}}}{{16l}}
Now, ll will be cancelled out on both sides.
⇒θ−θ02=θ016\Rightarrow \dfrac{{\theta - {\theta _0}}}{2} = \dfrac{{{\theta _0}}}{{16}}
⇒θ0=89θ\Rightarrow {\theta _0} = \dfrac{8}{9}\theta
Therefore, the twist angle at the joint will be 8θ9\dfrac{{8\theta }}{9}

So, the correct option is: (D) 8θ9\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.