A massless rope connects two blocks m1 = 4 kg on a frictionless table and m2 = 1 kg hanging over a pulley. What is the acceleration and the tension?

A massless rope connects two blocks m1 = 4 kg on a frictionless table and m2 = 1 kg hanging over a pulley. What is the acceleration and the tension?

• A. a ≈ 9.8 m/s^2; T ≈ 39.2 N
• B. a ≈ 1.96 m/s^2; T ≈ 7.84 N
• C. a ≈ 2.50 m/s^2; T ≈ 10.0 N
• D. a ≈ 0.98 m/s^2; T ≈ 3.92 N

Answer: (B)

This problem uses Newton's second law for a system of connected masses: the net external force on the system equals the total mass times the acceleration. Since the rope is massless and the pulley is frictionless, the tension is the same on both sides and there’s no friction on the table. For the block on the table, the only horizontal force is the tension, so T = m1 a. For the hanging block, downward weight minus upward tension equals its mass times acceleration: m2 g − T = m2 a. Combine the two: m2 g − m1 a = m2 a, so m2 g = (m1 + m2) a and a = m2 g /(m1 + m2) = (1 kg)(9.8 m/s^2)/(5 kg) = 1.96 m/s^2. The hanging mass accelerates downward, the table mass to the right. Tension then is T = m1 a = 4 kg × 1.96 m/s^2 = 7.84 N (also T = m2 g − m2 a = 9.8 − 1.96 = 7.84 N). Answer: acceleration ≈ 1.96 m/s^2; tension ≈ 7.84 N.

This problem uses Newton's second law for a system of connected masses: the net external force on the system equals the total mass times the acceleration. Since the rope is massless and the pulley is frictionless, the tension is the same on both sides and there’s no friction on the table.

For the block on the table, the only horizontal force is the tension, so T = m1 a.

For the hanging block, downward weight minus upward tension equals its mass times acceleration: m2 g − T = m2 a.

Combine the two: m2 g − m1 a = m2 a, so m2 g = (m1 + m2) a and a = m2 g /(m1 + m2) = (1 kg)(9.8 m/s^2)/(5 kg) = 1.96 m/s^2. The hanging mass accelerates downward, the table mass to the right.

Tension then is T = m1 a = 4 kg × 1.96 m/s^2 = 7.84 N (also T = m2 g − m2 a = 9.8 − 1.96 = 7.84 N).

Answer: acceleration ≈ 1.96 m/s^2; tension ≈ 7.84 N.