Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lecture 22

Quiz

This is a one-dimensional problem. Suppose a particle is attracted to the origin with a force

3x

kFx

Find the potential function.

Work-energy theorem:

12212121 KEKEWWW nccontotal

122112 KEKEWUU nc

112221 UKEUKEW nc

constIf 112221 ,0 UKEUKEW nc

Mechanical energy is conserved!

Examples

Strategy: write down the total mechanical energy, E,

E = KE + U at the initial and final positions of a particle:

Initial E1=KE1+U1…

Final E2=KE2+U2

Then use

or

112221 ,0 UKEUKEW nc If

112221 UKEUKEW nc

H

Water SlideWho hits the bottom with a faster speed?

Roller CoasterYou are in a roller coaster car of mass M that

starts at the top, height H, with an initial speed V0=0. Assume no friction.

a) What is the speed at the bottom?b) How high will it go again?

c) Would it go as high if there were friction?

H

Roller Coaster with FrictionA roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2).

Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

Conservative Forces

If there are only conservative forces in the problem, then there is conservation of mechanical energy

• Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another– Good examples: Gravity and Springs

• Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost.– Good example: Friction (like on Roller Coasters)

Law of Conservation of Energy• Mechanical Energy NOT always

conserved• If you’ve ever watched a roller

coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc.

• Energy = Kinetic Energy + Potential Energy + Heat + Others…–Total Energy is what is

conserved! K1+U1 = K2+U2+EHeat…

Total Energy is what is conserved! K1+U1= K2+U2+EHeat…

A gun shoots a bullet at angle θ with the x axis with a velocity of magnitude Vm. What is magnitude of the velocity when the bullet returns to the ground? How high will it go?