A cart of mass is initially at rest at the bottom of a frictionless track that ends…
PHYSICS · Multi-part · CBSE · Physics · Work, Energy and Power · Work, Energy and Power
Problem
A cart of mass 2.0kg is initially at rest at the bottom of a frictionless track that ends in a straight 30∘ incline, as shown in the side view description below. The cart is attached to a horizontal ideal spring of force constant k=400Nm−1 that is fixed to a wall located 0.20m before the base of the incline. The cart is pulled toward the wall, compressing the spring, and then released. As soon as the cart leaves the spring, it begins to move up the incline, where a constant coefficient of kinetic friction μk=0.10 acts over the entire incline surface. The goal is for the cart to travel a distance of 1.5m along the incline before momentarily coming to rest.
(a) On the outline of the cart provided below (students will reproduce or describe), draw and clearly label all the forces acting on the cart while it is ascending the incline.
(b) Write an algebraic relationship among the relevant energy terms and work done by non-conservative forces that applies from the moment the cart leaves the spring to the instant it comes to rest 1.5m up the incline. Do not substitute numbers yet.
(c) Derive an expression for the minimum spring compression xmin required to achieve the stated goal in terms of m, g, k, μk, the incline angle θ, and the distance d traveled along the incline.
(d) Calculate the numerical value of xmin using the given data.