# The outside diameter of a hollow shaft is thrice to its inside diameter. The ratio of its torque carrying capacity to that of a solid shaft of the sam

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The outside diameter of a hollow shaft is thrice to its inside diameter. The ratio of its torque carrying capacity to that of a solid shaft of the same material and the same outside diameter is:
1. 80/81
2. 1/81
3. 8/9
4. 1/9

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Correct Answer - Option 1 : 80/81

Concept:

The equation for Shaft Subjected to Torsion T

$\frac{τ }{R}$ = $\frac{T}{J}$ = $\frac{{Gθ }}{L}$

Solid shaft J = $\frac{{\pi D_s^4}}{{32}}$

Hollow Shaft J = $\frac{{\pi \left( {D_0^4 - \;D_i^4} \right)}}{{32}}$

τ = Shear stress induced due to torsion T, G = Modulus of Rigidity, θ = Angular deflection of the shaft, R, L = Shaft radius and length respectively, Ds = Diameter of the solid shaft, D0 = Outer Diameter of Hollow shaft, Di = Inner Diameter of Hollow shaft

Calculation;

Given

D0 = 3 × Di

Ds = D0  (Solid shaft diameter is equal to an outer diameter of hollow shaft)

Material is the same for the hollow and solid shaft (i.e. G is the same for both because G depends upon the material)

$\frac{T}{\tau }$ = $\frac{{G\theta }}{L}$

∴ T  $\propto \;$ J  (θ, G, L same for both or constant)

$\frac{{{T_H}}}{{{T_s}}}$ = $\frac{{{J_H}}}{{{J_s}}}$ = $\frac{{\frac{\pi }{{32}}\left( {D_0^4 - D_i^4} \right)}}{{\frac{\pi }{{32}}{{\left( {{D_s}} \right)}^4}}}$

Put, D0 = 3 × Di

Ds = D0

$J_H\over J_s$ = $80\over 81$

Polar moment of inertia

J =  $\mathop \smallint \limits_0^R 2\pi$r3dr

For Solid Shaft, J = $\pi D^4 \over 32$

For Hollow Shaft, J = $\mathop \smallint \limits_r^R 2\pi$r3dr = $\frac{{\pi \left( {D_0^4 - \;D_i^4} \right)}}{{32}}$

The assumption for Torsion equation:

$\tau\over R$ = $T\over J$ = $G\theta\over L$

• The bar is acted upon by pure torque.
• The section under consideration is remote from the point of application of the load and from a change in diameter.
• The material must obey Hooke's Law
•  Cross-sections rotate as if rigid, i.e. every diameter rotates through the same angle.