10. Conservation of Energy

FIGURE EX10.28 shows the potential-energy diagram for a 500 g particle as it moves along the x-axis. Suppose the particle's mechanical energy is 12 J. Where are the particle's turning points?

A particle with a mass of 0.1kg moves according to the Potential Energy graph shown. What minimum speed does the particle need at Point A to reach Point B?

A system in which only one particle moves has the potential energy shown in FIGURE EX10.31. What is the x-component of the force on the particle at x = 5, 15, and 25 cm?

In FIGURE EX10.28, what is the maximum speed a 200 g particle could have at x = 2.0 m and never reach x = 6.0 m?

FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0 m. What is the particle's maximum speed? At what position does it have this speed?

FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?

A clever engineer designs a 'sprong' that obeys the force law F_x=−q(x−x_eq)³ , where x_eq is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, we'll let x_eq = 0 m .Then F_x = −qx³. Find an expression for the potential energy of a stretched or compressed sprong.

A particle moving along the y-axis is in a system with potential energy U = 4y³ J, where y is in m. What is the y-component of the force on the particle at y = 0 m, 1 m, and 2 m?

A particle that can move along the x-axis is part of a system with potential energy U(x) = A/x² − B/x where A and B are positive constants. Where are the particle's equilibrium positions?

A system has potential energy $U\left(x\right)=\left(10\text{J}\right)[1-\sin (\left(3.14\text{rad/m}\right)x\left)\right]$ as a particle moves over the range $0\text{ m }\le x\le 3\text{ m}$. For each, is it a point of stable or unstable equilibrium?

A system consists of interacting objects A and B, each exerting a constant 3.0 N pull on the other. What is ∆U for the system if A moves 1.0 m toward B while B moves 2.0 m toward A?

A particle moving along the x-axis is in a system that has potential energy U = x³ - 3x J, where x is in m. For each, is it a point of stable or unstable equilibrium?

CALC The potential energy for a particle that can move along the x-axis is U = Ax² + B sin(πx/L), where A, B, and L are constants. What is the force on the particle at (a) x = 0, (b) x = L/2, and (c) x = L?

In FIGURE EX10.26, What minimum speed does a 100 g particle need at point B to reach point A?

In FIGURE EX10.27, what is the maximum speed of a 2.0 g particle that oscillates between x = 2.0 mm and x = 8.0 mm