Chapter 8. Potential Energy and

Energy Conservation

8.1. What is Physics?      8.2. Work and Potential Energy      8.3. Path Independence of Conservative Forces      8.4. Determining Potential Energy Values      8.5. Conservation of Mechanical Energy      8.6. Reading a Potential Energy Curve      8.7. Work Done on a System by an External Force      8.8. Conservation of Energy

Introduction

In Chapter 7 we introduced the concepts of work and kinetic energy. We then derived a net work-kinetic energy theorem to describe what happens to the kinetic energy of a single rigid object when work is done on it. In this chapter we will consider a systems composed of several objects that interact with one another.

What is Physics?  (1) The system consists of an Earth–

barbell system that has its arrangement changed when a weight lifter (outside of the system) pulls the barbell and the Earth apart by pulling up on the barbell with his arms and pushing down on the Earth with his feet

(2) The system consists of two crates and a floor. This system is rearranged by a person (again, outside the system) who pushes the crates apart by pushing on one crate with her back and the other with her feet

There is an obvious difference between these two situations. The work the weight lifter did has been stored in the new configuration of the Earth-barbell system, and the work done by the woman separating the crates seem to be lost rather than stored away.

How do we determine whether the work done by a particular type of force is “stored” or “used up.”?

The Path Independence Test for a Gravitational Force

The net work done on the skier as she travels down the ramp is given by

It does not depend on the shape of the ramp but only on the vertical component of the gravitational force and the initial and final positions of her center of mass.

Path Dependence of Work Done by a

Friction Force

• The work done by friction along that path 1→2 is given by

• The work done by the friction force along path 1→4→3→2 is given by

Conservative Forces and Path Independence

• conservative forces are the forces that do path independent work;

• Non-conservative forces are the forces that do path dependent work;

The work done by a conservative force along any closed path is zero.

Test of a System's Ability to Store Work Done by Internal Forces: the work done by a conservative internal force can be stored in the system as potential energy, and the work done by a non-conservative internal force will be “used up”

cU W

EXAMPLE 1: Cheese on a Track

Figure a shows a 2.0 kg block of slippery cheese that slides along a frictionless track from point 1 to point 2. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m. How much work is done on the cheese by the gravitational force during the slide?

Determining Potential Energy Values

Consider a particle-like object that is part of a system in which a conservative force acts. When that force does work W on the object,

the change in the potential energy associated with the system is the negative of the work done

Gravitational Potential Energy

GRAVITATIONAL POTENTIAL ENERGY

• The gravitational potential energy U is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level:

• SI Unit of Gravitational Potential Energy: joule (J)

GU mgh

Elastic Potential Energy

we choose the reference configuration to be when the spring is at its relaxed length and the block is at .

                                                                                                                      

                                 

or

Sample Problem 2 A 2.0 kg sloth hangs 5.0 m

above the ground (Fig. 8-6). • a) What is the gravitational

potential energy U of the sloth–Earth system if we take the reference point y=0 to be (1) at the ground, (2) at a balcony floor that is 3.0 m above the ground, (3) at the limb, and (4) 1.0 m above the limb? Take the gravitational potential energy to be zero at y=0.

• (b) The sloth drops to the ground. For each choice of reference point, what is the change in the potential energy of the sloth–Earth system due to the fall?

What is mechanical energy of a system?

The mechanical energy is the sum of kinetic energy and potential energies:

mec

sys sys sysE K U

For example,

2 21 1

2 2

mec

sys sys sysE K U mv mgh kx

Conservation of Mechanical Energy In a system where (1) no work is done on it by

external forces and (2) only conservative internal forces act on the system elements, then the internal forces in the system can cause energy to be transferred between kinetic energy and potential energy, but their sum, the mechanical energy Emec of the system, cannot change.

0mecsys sys sysE K U

An isolated system: is a system that there is no net work is done on the system by external forces.

Example 3

Check Your Understanding 

Some of the following situations are consistent with the principle of conservation of mechanical energy, and some are not. Which ones are consistent with the principle?

(a) An object moves uphill with an increasing speed. (b) An object moves uphill with a decreasing speed.(c) An object moves uphill with a constant speed.(d) An object moves downhill with an increasing speed. (e) An object moves downhill with a decreasing speed. (f) An object moves downhill with a constant speed.

Example 4  A Daredevil Motorcyclist

A motorcyclist is trying to leap across the canyon shown in Figure by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.

EXAMPLE 5: Bungee Jumper

A 61.0 kg bungee-cord jumper is on a bridge 45.0 m above a river. The elastic bungee cord has a relaxed length of L = 25.0 m. Assume that the cord obeys Hooke's law, with a spring constant of 160 N/m. If the jumper stops before reaching the water, what is the height h of her feet above the water at her lowest point?

EXAMPLE 6• In Fig., a 2.0 kg package of tamales slides along a floor

with speed v1=4.0 m/s. It then runs into and compresses a spring, until the package momentarily stops. Its path to the initially relaxed spring is frictionless, but as it compresses the spring, a kinetic frictional force from the floor, of magnitude 15 N, acts on it. The spring constant is 10 000 N/m. By what distance d is the spring compressed when the package stops?

Net Work on a system

Internal Work on a single rigid object

int 0W

Internal Work on a system

Since Newton's Third Law tells us that

the internal work is given by the integral of y

the internal work on a system is not zero in general

Work-Energy Theorem

intext mecNC sys sys sysW W K U E

Example 7  Fireworks• A 0.20-kg rocket in a fireworks

display is launched from rest and follows an erratic flight path to reach the point P, as Figure shows. Point P is 29 m above the starting point. In the process, 425 J of work is done on the rocket by the nonconservative force generated by the burning propellant. Ignoring air resistance and the mass lost due to the burning propellant, find the speed vf of the rocket at the point P.

Reading a Potential Energy Curve

Finding the Force Analytically

Solving for F(x) and passing to the differential limit yield

Reading a Potential Energy Curve • Turning Points: a place where K=0

(because U=E ) and the particle changes direction.

• Neutral equilibrium: the place where the particle has no kinetic energy and no force acts on it, and so it must be stationary.

• unstable equilibrium: a point at which . If the particle is located exactly there, the force on it is also zero, and the particle remains stationary. However, if it is displaced even slightly in either direction, a nonzero force pushes it farther in the same direction, and the particle continues to move

• stable equilibrium: a point where a particle cannot move left or right on its own because to do so would require a negative kinetic energy

Sample Problem A 2.00 kg particle moves along an x axis in one-dimensional

motion while a conservative force along that axis acts on it. The potential energy U(x) associated with the force is plotted in Fig. 8-10a. That is, if the particle were placed at any position between x=0 and x=7m , it would have the plotted value of U. At x=6.5m , the particle has velocity v0=(-4.0m/s)i . (a) determine the particle’s speed at x1=4.5m. (b) Where is the particle’s turning point located? (c) Evaluate the force acting on the particle when it is in the region 1.9m<x<4.0m.

General Energy Conservation

THE PRINCIPLE OF CONSERVATION OF ENERGY: Energy can neither be created nor destroyed, but can only be converted from one form to another.

int

int

int

ext mecNC sys sys sys

NCNC

ext mec mec NCsys NC sys

W W K U E

W E

W E W E E

For a isolated system where Wext is zero, it energy is conserved.

0mec NCsysE E

Example  In Fig. 8-58, a block slides along a path that is without

friction until the block reaches the section of length L=0.75m, which begins at height h=2.0m on a ramp of angle θ=30o . In that section, the coefficient of kinetic friction is 0.40. The block passes through point A with a speed of 8.0 m/s. If the block can reach point B (where the friction ends), what is its speed there, and if it cannot, what is its greatest height above A?