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Question

# Two shafts, one of solid section and the other of hollow section, of same material and weight having same length are subjected to equal torsional force. What is the torsional stiffness of hollow shaft?

A
Equal to that of the solid shaft
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B
Less than that of the solid shaft
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C
More than that of the solid shaft
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D
Exactly half of that of the solid shaft
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Solution

## The correct option is C More than that of the solid shaftLet the diameter of solid shaft be ds. Let the external and internal diameter of the hollow shaft be do and di respectively. It is given that both the shafts are made of same material, have same weight and length and are subjected to equal torsional force. ∴ Weight of solid shaft = Weight of hollow shaft ⇒ γ×π4d2sL=γ×π4(d2o−d2i)L ⇒ d2s=d2o−d2i .....(i) Now, we know that, torsional stiffness is given as k=Tθ=GIPL For same value of G and L k=Tθ∝Ip ⇒ khks=π32(d4o−d4i)π32(d4s) ⇒ khks=(d2o)2−(d2i)2(d2s)2 ⇒ khks=(d2o−d2i)(d2o+d2i)(d2s)2 [from(i)] ⇒ khks=d20+d2id2o−d2i Now,the quantityd2o+d2id2o−d2iis always greater than 1 ∴ kh>ks

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