MHT CET · Physics · Laws of Motion

An elevator of mass $m$ is connected to a rope which pulls the elevator up with a maximum acceleration equal to $a$ . If the maximum bearable tension in the rope is $T$ , then for a safe journey the minimum diameter of the rope is [ $g$ = acceleration due to gravity]

A ${[\frac{2 m (g + a)}{π T}]}^{\frac{1}{2}}$
B ${[\frac{4 m (g + a)}{π T}]}^{\frac{1}{2}}$
C ${[\frac{m (g + a)}{π T}]}^{\frac{1}{2}}$
D ${[\frac{m (g + a)}{2 π T}]}^{\frac{1}{2}}$

Verified Solution

Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

The maximum tension in the rope

= m (g + a)

Stress in the rope

= \frac{m (g + a)}{π r^{2}}

∴ T = \frac{m (g + a)}{π r^{2}} = \frac{m (g + a)}{π {(\frac{d}{2})}^{2}}

∴ T = \frac{4 m (g + a)}{π d^{2}}

∴ d^{2} = \frac{4 m (g + a)}{π T}

∴ d = {[\frac{4 m (g + a)}{π T}]}^{\frac{1}{2}}

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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An elevator of mass m is connected to a rope which pulls the elevator up with a maximum acceleration equal to a . If the maximum bearable tension in the rope is T , then for a safe journey the minimum diameter of the rope is [ g = acceleration due to gravity]

Step-by-step Solution

An elevator of mass $m$ is connected to a rope which pulls the elevator up with a maximum acceleration equal to $a$ . If the maximum bearable tension in the rope is $T$ , then for a safe journey the minimum diameter of the rope is [ $g$ = acceleration due to gravity]