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PY 11: GATE ME 2016 Official Paper: Shift 3

Option 4 : θ_{s} = θ_{H} and τ_{s} < τ_{H}

CT 1: Ratio and Proportion

2672

10 Questions
16 Marks
30 Mins

**Concept:**

\({\tau _{max}}\;for\;solid\;shaft = \frac{T}{J}.r\)

\({\tau _{max}}\;for\;hollow\;shaft = \frac{T}{J}.{r_0}\)

where T = Torque, J = Polar Area Moment, r = radius of solid shaft and r_{0} = outer radius of hollow shaft

**Calculation:**

**Given:-**

**L _{S} = L_{H}, T_{S} = T_{H} = T, J_{S} = J_{H} **

(Where L = length, T = torque, J = Polar moment of inertia)

**∵ J _{S} = J_{H}**

\(\frac{{\pi {d^4}}}{{32}} = \frac{\pi }{{32}}\left[ {d_0^4 - d_i^4} \right]\)

\(\Rightarrow {d_o} > d \Rightarrow {r_0} > r\)

**Now, angle of Twist**

\(\theta = \frac{{TL}}{{GJ}}\)

∴ **θ _{S} = θ_{H}**

\(\tau = \frac{{TR}}{J}\)

\(\Rightarrow {\tau _{max}}\;for\;solid\;shaft = \frac{T}{J}.r\)

\({\tau _{max}}\;for\;hollow\;shaft = \frac{T}{J}.{r_0}\)

∵ r_{0} > r

**∴**** (τ _{max})_{H} > (τ_{max})_{S } ----(2)**