PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Work Energy and Power · Work Energy and Power
Problem
A cart of mass m sits on a horizontal track. A light horizontal spring of force constant k is attached to the front of the cart. A constant horizontal force of magnitude F is applied to a rigid hook fixed to the opposite (rear) side of the cart, pushing the cart and simultaneously compressing the spring as both move to the right. During the push the hook remains firmly attached to the cart, so the force acts at the same point on the cart for the entire motion. Friction is negligible.
Initially the spring is relaxed and the cart is at rest. The force acts until the spring momentarily comes to rest after being compressed a maximum distance xmax.
(a) Taking the cart alone as the system, write an energy equation from the start of the push until the instant of maximum compression. Identify each term.
(b) Now consider the cart + spring together as a single system. Write an energy equation for this larger system over the same interval and explain how it differs from the equation in part (a).
(c) Derive an expression for xmax in terms of F and k. Show all steps clearly.
(d) Suppose the applied force is doubled while all other quantities remain the same. Predict qualitatively how xmax changes and justify your answer using your result from part (c).