MHT CET · Physics · Mechanical Properties of Solids

A lift is tied with thick iron ropes having mass ‘M’. The maximum acceleration of the lift is ‘a’ $m / s^{2}$ and maximum safe stress is ‘S’ $N / m^{2}$ . The minimum diameter of the rope is

A ${[\frac{6 M (g + a)}{π S}]}^{\frac{1}{2}}$
B ${[\frac{4 M (g + a)}{π S}]}^{\frac{1}{2}}$
C ${[\frac{M (g + a)}{π S}]}^{\frac{1}{2}}$
D ${[\frac{M (g - a)}{π S}]}^{\frac{1}{2}}$

Verified Solution

Answer & Solution

Correct Answer

(B) ${[\frac{4 M (g + a)}{π S}]}^{\frac{1}{2}}$

Step-by-step Solution

Detailed explanation

The maximum stress produced in a rope is given by

σ_{m a x} = \frac{F o r c e}{A r e a} = \frac{M g}{π r^{2}}

As the lift is accelerating with acceleration a, then

σ_{m a x} = \frac{M (g \pm a)}{π r^{2}}

\Rightarrow r^{2} = \frac{M (g \pm a)}{π S}

[Given,

σ_{m a x} = S]

\frac{d^{2}}{4} = \frac{M (g \pm a)}{π S} [∵ r = \frac{d}{2}]

\Rightarrow d = \sqrt{\frac{4 M (g \pm a)}{π S}}

As acceleration is maximum i.e.’=g +a, so

d = \sqrt{\frac{4 M (g + a)}{π S}}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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A lift is tied with thick iron ropes having mass ‘M’. The maximum acceleration of the lift is ‘a’ m/s2 and maximum safe stress is ‘S’ N/m2 . The minimum diameter of the rope is

Step-by-step Solution

A lift is tied with thick iron ropes having mass ‘M’. The maximum acceleration of the lift is ‘a’ $m / s^{2}$ and maximum safe stress is ‘S’ $N / m^{2}$ . The minimum diameter of the rope is