Which equation expresses the forces on a mass m suspended by a rope that accelerates downward with acceleration a?

• A. The equation m g − T = m a.
• B. The equation T − m g = m a.
• C. The equation m g + T = m a.
• D. The equation T = m g − m a.

Answer: (A)

When a mass on a rope accelerates downward, gravity pulls it down with a force m g, while the rope tension pulls upward with force T. Use Newton’s second law by choosing the downward direction as positive. The net force in that direction is m g (down) minus T (up), which must equal m times the downward acceleration a. So the equation is m g − T = m a. This form directly shows how the downward forces overcome tension to produce the downward acceleration. If you rearrange it, you can write T = m g − m a, which is the same relationship written differently. The other expressions would mix the directions of the forces or assign signs inconsistent with downward acceleration.