Chapter 8: Conservation of Energy

In Ch. 7, we learned

• The Work-Energy Principle: Wnet = (½)m(v2)2 - (½)m(v1)2 K Wnet ≡ The TOTAL work done by ALL forces!

• Kinetic Energy: K (½)mv2

• Work (constant force):

W = F||d =Fd cosθ

Sect. 8-1: Conservative & Nonconservative Forces

Definition: A force is conservative if & only ifthe work done by that force on an object moving from one point to another depends ONLY on the initial & final positions of the object, & is independent of the particular path taken. Example: gravity.

Conservative Force: Another definition:

A force is conservative if the net work done by the force on an object moving around any closed path is zero.

If friction is present, the work done depends not only on the starting & ending points, but also on the path taken.

Friction is a Nonconservative Force!

Friction is a Nonconservative Force.The work done by friction depends on the path!

Sect. 8-2: Potential Energy

A mass can have a Potential Energy due to its environment

Potential Energy (U)

The energy associated with the position or configuration of a mass.

Examples of potential energy:

A wound-up spring

A stretched elastic band

An object at some height above the ground

Potential Energy:

Can only be defined for

Conservative Forces!

• Potential Energy (U) Energy associated with the position or configuration of a mass.

Potential work done!

Gravitational Potential Energy:

Ugrav mgy

y = distance above Earth

m has the potential to do work

mgy when it falls

(W = Fy, F = mg)

So we Define the Gravitational Potential Energy at height y above some reference point as

In raising a mass m to a height h, the work done by the external force is

Gravitational Potential Energy

• Consider a problem in which the height of a mass above the Earth changes from y1 to y2:

• The Change in Gravitational Potential Energy is:

Ugrav = mg(y2 - y1)

• The work done on the mass by gravity is: W = Ugrav

y = distance above Earth

Where we choose y = 0 is arbitrary, since we take

the difference in 2 y’s in calculating Ugrav

Of course, this potential energy will be converted to kinetic energy if the object is dropped.

Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces).

If Ugrav = mgy, from where do we measure y?

Doesn’t matter, but we need to be consistent about this choice!

This is because only changes in potential energy can be measured.

Example 8-1: Potential energy changes for a roller coasterA roller-coaster car, mass m = 1000 kg, moves from point 1 to point 2 & then to point 3.

a. Calculate the gravitational potential energy at points 2 & 3 relative to a point 1. (That is, take y = 0 at point 1.)

b. Calculate the change in potential energy when the car goes from aa point 2 to point 3.

c. Repeat parts a. & b., but take the reference point (y = 0) at point 3.

∆U = mg∆yDepends only

on differences ∆yin vertical height!

A General Definition of gravitational potential energy

For any conservative force F:

Consider Again an Ideal Spring ForceOther types of potential energy besides

Gravitational exist. We can define a potential

energy for any conservative force. Recall (Ch. 7)

the ideal spring, characterized by a spring

constant k, a measure of spring “stiffness”.

Restoring force of spring acting on the hand:

Fs = -kx (Fs >0, x <0; Fs <0, x >0)

known as Hooke’s “Law” (but isn’t really a law!)

In Ch. 7, we showed that the work done by

the person is

W = (½)kx2 Ue (The definition of Elastic Potential Energy!!)

A spring has a potential energy, called elastic potential energy, when it is compressed or stretched. As we’ve said, the force required to compress or stretch a spring is:

where k is the spring constant. The potential energy is then:

Elastic Potential Energy

Relaxed Spring

Elastic Potential Energy, Ue = (½)kx2

Work to compress the spring a distance x:

W = (½)kx2 Ue The spring stores potential energy.

When the spring is released, it transfersit’s potential energy Ue = (½)kx2 to the mass in the form of kinetic energy K = (½)mv2

Elastic Potential Energy, Ue = (½)kx2

U1 = (½)kx2 K1 = 0

U2 = 0

K2 = (½)mv2

K1 + U1 = K2 + U2

• In a problem in which the compression or stretching distance of a spring changes from x1 to x2.

• The change in U is:

Uelastic = (½)k(x2)2 - (½)k(x1)2 • The work done is:

W = - Uelastic

The potential energy belongs to the system, not to individual objects

In general, given the potential energy U, we can formally find the force F as follows:

We can formally invert this equation to find F(x) if we know U(x):

In three dimensions this has the form: