Lecture 15

Summary

Springs
Conservative & nonconservative forces
Kinetic energy

Today's joke

Potential energy U

gravitational potential energy applet
I have not yet found an HTML5 equivalent to this applet.

Example #1

removing the constraint
energy conversion
spring potential energy applet
BU mass on spring energy applet
This applet is similar to the above except it draws an energy vs. time graph instead of an animated bar chart.

Conservative forces

independent of path
W_{closed path} = 0
examples: gravity, spring, electric forces
a conservative force internal to the system is associated with potential energy:
W_int = −ΔU

Nonconservative forces

path-dependent
W_{closed path} ≠ 0
examples: friction, engine forces

Energy diagrams

projectile applet1
BU rocket landing energy applet
This applet resembles the above except the ball in freefall suddenly turns on a rocket engine halfway down, which represents external work on the ball-Earth system. The first half does show how a freefalling ball trades gravitational potential energy for kinetic energy.

projectile applet2
BU mass on ramp energy applet
This applet resembles the above except it draws energy vs. time graphs (not bar charts) and it depicts an object sliding down a ramp instead of a projectile in freefall.

mass on horizontal spring applet
BU mass on spring energy applet
This applet is similar to the above except it only draws an energy vs. time graph and not an animated bar chart.

mass on vertical spring
Force is the −slope of U vs. x plot
stable & unstable equilibria

Example #2

Example #3

pse6 8.21
A 4.00-kg particle moves directly from (0, 0) to (4.00 m, 8.00 m) against gravity acting in the −y direction. What is ΔU_g?
A. 314 J
B. −351 J
C. 32 J
D. −112 J
Answer

Knight2 Fig 10-31 klm
A ball is fired upward at (a), rises halfway at (b), reaches maximum height at (c), and falls one quarter of the way back at (d). Which of the following sets of energy diagrams correspond correctly to the situation?
ball rising and falling

Answer

Knight2 Fig 10-34 klm
A particle with the indicated total energy TE is released at x = 0 and is subject to a potential energy U indicated by the blue curve. At which point does the particle achieve its maximum speed?
particle subject to potential energy curve

particle subject to potential energy curve

Answer

answer

A. 314 J
solution equation

The kinetic energy is maximum at launch, decreases to zero at the peak of the ball's flight, then increases as it falls back toward Earth.

C. answer to particle subject to potential energy curve question

answer to particle subject to potential energy curve question

The kinetic energy of the particle (and hence its speed) will be greatest when its potential energy is least.

monster truck joke

This cartoon caption was written by the famous movie critic Roger Ebert. It was not a winning entry, but I like it. I saw the cartoon in an online article after his death. And here is a blog written by Mr. Ebert about his cartoon writing efforts.