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Question

A shaft is fixed at one end and free at another end. It carries uniformly distributed torque (t kN - m/m) as shown in the figure. The shaft is prismatic and have constant GJ. The angle of twist at the free end is:

A
tL28GJ
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B
tL22GJ
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C
3tL28GJ
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D
None of these
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Solution

The correct option is C 3tL28GJ
We know, TJ=GθLdθ=TxdxGJ
Now, Tx=⎪ ⎪⎪ ⎪tx 0<x<L2tL2 L2<x<L
θ=L/20txdxGJ+LL/2(tL2)dxGJ
=tGJ(x22)L/20+(tL2)1GJ(LL2)
=t2GJ(L24)+tL2GJ×L2
=tL28GJ+tL24GJ=3tL28GJ

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