Question

Two solid shafts ‘A’ and ‘B’ are made of the same material. The shaft ‘A’ is 50 mm diameter and shaft ‘B’ is 100 mm diameter. The ratio of strength of shaft ‘B’ compared to shaft ‘A’ is

1. \(\frac{1}{2}\)
2. 2
3. 4
4.

8

Answer 1

Correct Answer - Option 4 :

8

Concept:

Safe Condition for design w.r.t strength criterion i.e., Torque carrying capacity

τ_max ≤ τ _per

\(\frac{T}{Z_p}\) ≤ τ _per

T ≤ Z_pτ per ...(i)

for safe condition

T ≤ T_R ...(ii)

Where T = applied torque, T_R = resisting torque, τ_per = Permissible stress, τ_max = Maximum stress, Z_p = Polar section modulus

Comparing (i) and (ii) 

T_R = Z_pτ_per

For two shafts made up of the same material, their strength is directly proportional to the polar section modulus. 

\({Z_p} = \frac{{{I_p}}}{r} = \frac{J}{r} = \frac{{\frac{{\pi {D^4}}}{{32}}}}{{\frac{D}{2}}} = \frac{{\pi {D^3}}}{{16}}\)

∴ Z_p ∝ D³

Calculation:

Given:

DA = 50 mm, D_B = 100 mm

\(\frac{{{Z_{pB}}}}{{{Z_{pA}}}} = {\left( {\frac{{{D_{B}}}}{{{D_{A}}}}} \right)^3} =(\frac{100}{50})^3= {\left( 2 \right)^3} = 8\)

Z_pB = 8 × Z_pA

Therefore, The ratio of the strength of shaft ‘B’ compared to shaft ‘A’ is 8 times