A cart of mass 0.50 kg is attached to an ideal horizontal spring on a nearly friction-free air track. The cart is displaced 0.080 m to the right of the equilibrium position and released from rest at time t=0. Measurements show that one full oscillation requires 0.90 s.
Answer the following questions. Neglect all forms of friction.
(1) Determine a mathematical expression for the cart’s displacement x(t), with numerical values for the amplitude and angular frequency, that satisfies the stated initial conditions.
(2) Obtain explicit expressions for the cart’s velocity v(t) and acceleration a(t) as functions of time.
(3) Calculate the maximum speed and the maximum magnitude of acceleration attained by the cart during its motion.
(4) Find the displacement, velocity, and acceleration of the cart at t=0.225s. Give each answer with appropriate units and signs.
(5) Using your results from part (d), explain how the directions of the displacement and acceleration at t=0.225s are consistent with the nature of the restoring force exerted by the spring.