Elevator dynamics — Cable tension comparison:\nIn which case is the tension in the cable supporting a lift (elevator) greater: when the lift is moving upwards with acceleration, or when it is moving downwards with acceleration?

Introduction / Context:
Elevator (lift) problems are standard applications of Newton's second law. Comparing cable tension during accelerated motion is important for sizing hoists, motors, and safety brakes. The direction of acceleration relative to weight determines whether the cable must supply extra force or is partially relieved.

Given Data / Assumptions:

• Lift of mass m moving vertically.
• Acceleration magnitude = a; direction either upward or downward.
• Gravity acts downward with magnitude W = m * g.
• Cable mass and pulley friction neglected for conceptual comparison.

Concept / Approach:
Apply ΣF = m * a along the vertical direction. Tension T counters weight and provides acceleration. The sign of a changes the required T. Upward acceleration requires T > W; downward acceleration requires T < W (assuming nonzero a).

Step-by-Step Solution:
Case 1 (accelerating upward): Take upward positive. ΣF = T − W = m * a ⇒ T = W + m * a = m * (g + a).Case 2 (accelerating downward): Take upward positive. ΣF = T − W = −m * a ⇒ T = W − m * a = m * (g − a).Compare: m * (g + a) > m * (g − a) for a > 0.Therefore, tension is greater when the lift accelerates upward.

Verification / Alternative check:
If a = 0 (uniform speed), T = W in both directions. As a increases upward, T increases above W; as a increases downward, T decreases below W but remains positive as long as a < g, which is typical in normal operation.

Why Other Options Are Wrong:

• downwards: Would imply T > W − m a exceeds T for upward case, which contradicts ΣF = m a.
• same in both cases: Only true for a = 0 (not our accelerated scenario).
• indeterminate: Laws of motion determine this unambiguously.

Common Pitfalls:

• Confusing the sign convention, leading to T = m * (g − a) for upward motion (incorrect).
• Thinking speed (not acceleration) changes tension; constant-speed motion does not alter T from W.

Final Answer:
upwards