Question

If two aluminum bar have a different length (L₁ = 2L₂) and diameter (d₁ = 2d₂) with an identical angle of a twist then, find torque value for bar 1, If bar 2 torque value is 50 N-m

1. 6.25 N-m
2. 200 N-m
3. 12.5 N-m
4. 400 N-m

Answer 1

Correct Answer - Option 4 : 400 N-m

Concept:
We know that the torsion equation is:

\(\frac{{{\tau _{max}}}}{R} = \frac{{G\theta }}{L} = \frac{T}{J}\)

where θ = the angle of twist, T = torque applied, L = length of the shaft, J = polar moment of inertia = \(J = \frac{{\pi {d^4}}}{{32}}\), d = diameter of the shaft

Calculation:

Given:
d1 = 2d₂, L1 = 2L₂, θ₁ = θ₂, T₂ = 50 N-m

\( {T}=\frac{{JG\theta }}{L} = \frac{{\pi}{d^4}{G\theta }}{32L}\)

\(\theta =\frac{{32TL}}{{\pi}{d^4}{G}}\)

For two aluminum bar having an equal angle of twist the equation can be written as,

\(\frac{{T_1L_1}}{{}{d_1^4}} =\frac{{T_2L_2}}{{}{d_2^4}}\)

\({T_1} =\frac{{T_2L_2{d_1^4}}}{{}{d_2^4}{L_1}} =\frac{{50\times L_2{(2d_2)^4}}}{{}{d_2^4}\times{2L_2}}=\frac{50\times16}{2}= 400\; N.m\)