In a frictionless pulley system with a 6 kg hanging mass and a 3 kg block on a horizontal table connected by a rope over a massless pulley, what is the acceleration of the block on the table?

In a frictionless pulley system with a 6 kg hanging mass and a 3 kg block on a horizontal table connected by a rope over a massless pulley, what is the acceleration of the block on the table?

• A. 4.87 m/s^2
• B. 6.53 m/s^2
• C. 2.18 m/s^2
• D. 9.8 m/s^2

Answer: (B)

In this setup, the rope and pulley are ideal, so the tension is the same on both sides and the acceleration is the same for both masses. Apply Newton’s second law to each mass and use the constraint that their accelerations have the same magnitude but opposite directions. For the 6 kg hanging mass, gravity pulls it down and tension pulls it up: m_h g − T = m_h a. For the 3 kg mass on the table, the only horizontal force is the tension pulling it toward the pulley: T = m_t a. Substitute T from the second equation into the first: m_h g − m_t a = m_h a, which gives m_h g = (m_h + m_t) a, so a = m_h g / (m_h + m_t). Plugging in m_h = 6 kg and m_t = 3 kg and g ≈ 9.8 m/s^2: a = 6 × 9.8 / (6 + 3) = 58.8 / 9 ≈ 6.53 m/s^2. So the block on the table accelerates to the right at about 6.53 m/s^2 (the hanging mass accelerates downward with the same magnitude).

In this setup, the rope and pulley are ideal, so the tension is the same on both sides and the acceleration is the same for both masses. Apply Newton’s second law to each mass and use the constraint that their accelerations have the same magnitude but opposite directions.

For the 6 kg hanging mass, gravity pulls it down and tension pulls it up: m_h g − T = m_h a.

For the 3 kg mass on the table, the only horizontal force is the tension pulling it toward the pulley: T = m_t a.

Substitute T from the second equation into the first: m_h g − m_t a = m_h a, which gives m_h g = (m_h + m_t) a, so a = m_h g / (m_h + m_t).

Plugging in m_h = 6 kg and m_t = 3 kg and g ≈ 9.8 m/s^2:

a = 6 × 9.8 / (6 + 3) = 58.8 / 9 ≈ 6.53 m/s^2.

So the block on the table accelerates to the right at about 6.53 m/s^2 (the hanging mass accelerates downward with the same magnitude).