Question

If the tension in the cable supporting a lift moving downwards is half the tension when it is moving upwards, the acceleration of the lift is

• A. \( \dfrac{g}{2} \)
• B. \( \dfrac{g}{3} \)
• C. \( \dfrac{g}{4} \)
• D. none of these

Hint:

For lifts: Upward acceleration → \( T = m(g + a) \)
Downward acceleration → \( T = m(g - a) \)

Answer 1

The Correct Option is B

Step 1: For a lift of mass \( m \) accelerating at rate \( a \), the cable tension moving upward is \( T_1 = m(g+a) \) and moving downward is \( T_2 = m(g-a) \).
Step 2: The condition given is \( T_2 = \tfrac{1}{2}T_1 \), so \( m(g-a) = \tfrac{1}{2}m(g+a) \); cancel \( m \) and multiply both sides by \( 2 \) to get \( 2(g-a) = g+a \).
Step 3: Expanding gives \( 2g - 2a = g + a \), so \( g = 3a \). \[ \boxed{a = \dfrac{g}{3}} \]

Answer 2

Another way to reach the answer is to work directly with the ratio \( T_2/T_1 = 1/2 \) as a single equation in \( a/g \), rather than solving for tensions first, and check each option against that ratio equation.

1. Option \( g/2 \): Writing the ratio equation as \( \dfrac{g-a}{g+a} = \dfrac{1}{2} \) and substituting \( a/g = 1/2 \) gives \( \dfrac{1 - 0.5}{1+0.5} = \dfrac{0.5}{1.5} = \dfrac{1}{3} \ne \dfrac{1}{2} \), so this fails the ratio test.
2. Option \( g/3 \): Substituting \( a/g = 1/3 \) into \( \dfrac{g-a}{g+a} \) gives \( \dfrac{1-1/3}{1+1/3} = \dfrac{2/3}{4/3} = \dfrac{1}{2} \), which exactly satisfies the required ratio.
3. Option \( g/4 \): Substituting \( a/g = 1/4 \) gives \( \dfrac{1-1/4}{1+1/4} = \dfrac{3/4}{5/4} = \dfrac{3}{5} \ne \dfrac{1}{2} \), so this also fails.
4. Option "none of these": Since \( a = g/3 \) satisfies the ratio equation exactly, a valid answer is present among the listed choices, making this option unnecessary.

Rewriting the condition as a single ratio equation in \( a/g \) and testing each candidate value confirms the same acceleration found through direct substitution.

So the correct answer is \( g/3 \).