At theta = theta_max, which statement is true?

At theta = theta_max, which statement is true?

• A. F_restoring = 0; a_t = 0; U = 0; KE = max; v = max
• B. F_restoring = max; a_t = max; U = max; KE = 0; v = 0
• C. F_restoring = max; a_t = 0; U = max; KE = max; v = 0
• D. F_restoring = 0; a_t = max; U = max; KE = 0; v = 0

Answer: (B)

At the furthest point of the swing, the pendulum is momentarily at rest, so the speed is zero and the kinetic energy is zero. That means all the energy is stored as gravitational potential energy, which is largest at the top of the arc, so U is maximum. The restoring force that pulls the pendulum back toward the center is proportional to sin theta; since sin theta is largest at theta_max (for angles up to the maximum excursion), the restoring force has its greatest magnitude there. The tangential acceleration is this restoring force divided by the mass, so its magnitude is also maximal at theta_max. Putting it together: the restoring force is at maximum, the tangential acceleration is at maximum, the potential energy is at maximum, and both kinetic energy and speed are zero.

At the furthest point of the swing, the pendulum is momentarily at rest, so the speed is zero and the kinetic energy is zero. That means all the energy is stored as gravitational potential energy, which is largest at the top of the arc, so U is maximum. The restoring force that pulls the pendulum back toward the center is proportional to sin theta; since sin theta is largest at theta_max (for angles up to the maximum excursion), the restoring force has its greatest magnitude there. The tangential acceleration is this restoring force divided by the mass, so its magnitude is also maximal at theta_max. Putting it together: the restoring force is at maximum, the tangential acceleration is at maximum, the potential energy is at maximum, and both kinetic energy and speed are zero.