Welcome back to Physics 211

1
Welcome back to Physics 211

• Todays agenda
• Mechanical energy
• Conservation of energy

2
Reminder

• Exam 2 in class on Thursday
• Seating will be posted. Closed book
• Material forces, Newtons laws, work, power,
  kinetic and potential energy (not oscillations)
• Practice tests, formula summary online
• MPHW4 due Friday 11 pm

3
Defining gravitational potential energy
The change in gravitational potential energy of
the object-earth system is just another name for
the negative value of the work done on an object
by the earth.
4
Total energy for object moving under gravity

• Definition of gravitational potential energy
  allows us to write down a conservation law for
  the total energy
• D (KUg)0 gt EUgKconstant
• ½ mv2mgh constant
• Crucial that U only depends on height!

5
demo

• Ball rolling on inclined slope equal heights
  from conservation of energy

6
Conservative forces

• Potential energies can only be defined for
  conservative forces
• Those forces which do work on a object in moving
  it from one position to another in such a way
  that the work done is path independent.
• eg. gravity, electric forces, spring forces
• Not friction, air resistance

7
Pendulum demo

• Energy (KU) should be constant
• If pendulum released with zero speed will return
  to same point (height) with zero speed (ignoring
  air drag, friction etc)

8
Another example springs

• Force F-kx (Hookes law)
• (x extension)
• ? W-1/2kx2 (last time)
• Therefore, can define elastic (spring)
  potential energy
• U 1/2kx2

9

• As a spring connected to a mass is extended
  beyond its natural length the work done on the
  mass is negative. What is the sign of the work
  done on the mass when the spring is compressed ?
• positive
• negative
• zero
• it depends on which way the mass is moving

10
(Horizontal) Spring
x
frictionless table
spring
1/2kx21/2mv2constant
11
Motion ?

• Imagine displacing mass from equilibrium to xa
  what happens ?
• 1/2ka21/2kx21/2mv2 or
• 1/2mv2 1/2ka2-1/2kx2
• what range does x lie in ?
• when is v greatest ?
• describe motion

12
Graph
oscillates! motion confined to region below
dotted line
u(x)1/2kx2
a
-a
x
13
Force as slope of potential energy graph
Definition DU-FDx or F - DU/Dx Force
proportional to slope of U(x) curve Equilibrium
corresponds to F0 ie zero slope
14
Potential Energy graphs
unstable equilibrium
U(x)
E
stable equilibrium
true equilibrium
x
15
Many forces

• For a particle which is subject to several
  (conservative) forces F1, F2
• E1/2mv2U1U2 is constant
• eg. vertical spring gravity and spring forces
  operate simultaneously

16

• A compressed spring fires a ping pong ball
  vertically upward. If the spring is compressed by
  1 cm initially the ball reaches a height of 2 m
  above the spring. What height would the ball
  reach if the spring were compressed by just 0.5
  cm ?
• (neglect air resistance)
• 2 m
• 1 m
• 0.5 m
• we do not have sufficient information to
  calculate the new height

17
Solution

• use conservation of total mechanical energy ?

18
Many particles

• Eg Java billiard ball simulation
• Only sum of energies for all particles is
  conserved
• EE1E2constant

19
Summary

• Total (mechanical) energy of an isolated system
  is constant in time.
• Must be no non-conservative forces
• Must sum over all conservative forces
• Must sum over all particles making up system