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Today’s Topics

Potential Energy, Ch. 8-1

Conservative Forces, Ch. 8-2

Conservation of mechanical energy Ch.8-4
Physics 151: Lecture 15, Pg 1
New Topic - Potential Energy

Consider a ball at some height above the ground.
No Velocity
Some Velocity
Physics 151: Lecture 15, Pg 2
New Topic - Potential Energy

Consider a ball at some height above the ground.
What work is done in this process ?
(Work done by the earth on the ball)
h
W = F. Dx
W = mgh cos(0)
W = mgh
Physics 151: Lecture 15, Pg 3
New Topic - Potential Energy

Consider a ball at some height above the ground.
h
Before the ball falls it has the potential to
do an amount of work mgh.
We say the ball has a potential energy
of U = mgh.
By falling the ball loses its potential
energy, work is done on the ball, and it
gains some kinetic energy,
W = DK = 1/2 mv2 = -DU = mgh
Physics 151: Lecture 15, Pg 4

Lecture 15, ACT 1
Work Done by Gravity
The air track is at an angle of 30 degrees with respect
to horizontal. The cart (with mass 1 kg) is released. It
bounces back and forth on the track. It falls 1 meter
down the track, then bounces back up to its original
position. How much total work is done by gravity on the
cart when it reaches its original position.
30 degrees
A) 5 J
B) 10 J
C) 20 J
D) 0 J
Physics 151: Lecture 15, Pg 5
Some Definitions

Conservative Forces - those forces for which the
work done does not depend on the path taken,
but only the initial and final position.

Potential Energy - describes the amount of work
that can potentially be done by one object on
another under the influence of a conservative
force
W = -DU
only differences in potential energy matter.
Physics 151: Lecture 15, Pg 6
See text: 8.1
Potential Energy

For any conservative force F we can define a
potential energy function U in the following way:
W =
 F.dr = - DU
 The
work done by a conservative force is equal and
opposite to the change in the potential energy function.
r2

U2
This can be written as:
 F.dr
r2
DU = U2 - U1 = - W = -
r1
r1
U1
Physics 151: Lecture 15, Pg 7
Question - 1

For a force to be a conservative force, when
applied to a single test body:
a. it must have the same value at all points in
space.
b. it must have the same direction at all points in
space.
c. it must be parallel to a displacement in any
direction.
d. equal work must be done in equal
displacements.
e. no work must be done for motion in closed
paths.
Physics 151: Lecture 15, Pg 8
See text: 8-1
A Conservative Force : Spring

For a spring we know that Fx = -kx.
F(x)
x1
x2
x
relaxed position
-kx
F = - k x1
F = - k x2
See Figure 7-7
Physics 151: Lecture 15, Pg 9
See text: 8-1
Spring...

The work done by the spring Ws during a displacement
from x1 to x2 is the area under the F(x) vs x plot between
x
x1 and x2.
2
Ws   F ( x)dx
x1
x2
F(x)
x1
  ( kx)dx
x2
x1
x
Ws
-kx
1 2 x
  kx
x
2
1
Ws   k x22  x12 
2
2
1
1
U  Ws  k x22  x12 
2
See Figure 7-7
Physics 151: Lecture 15, Pg 10
Question - 2

The force a spring exerts on a body is a
conservative force because :
1.
a spring always exerts a force opposite to the
displacement of the body.
the work a spring does on a body is equal for
compressions and extensions of equal magnitude.
the work a spring does on a body is equal and
opposite for compressions and extensions of
equal magnitude.
the net work a spring does on a body is zero
when the body returns to its initial position.
2.
3.
4.
Physics 151: Lecture 15, Pg 11

Lecture 15, ACT 2
Work/Energy for Conservative Forces
The air track is is again at an angle of 30 degrees
with respect to horizontal. The cart (with mass 1 kg)
is released 1 meter from the bottom and hits the
bumper with some speed, v1. You want the cart to go
faster, so you release the cart higher. How high do
you have to release the cart so it hits the bumper with
speed v2 = 2v1?
30 degrees
A) 1 m
B) 2 m
C) 4 m
D) 8 m
Physics 151: Lecture 15, Pg 12
See text: 8-4
Conservation of Energy

If only conservative forces are present, the total energy
(sum of potential and kinetic energies) of a system is
conserved.
E=K+U
DE = DK + DU
= W + DU
= W + (-W) = 0
E = K + U is constant !!!
using DK = W
using DU = -W
Animation_2
Animation_3
Both K and U can change, but E = K + U remains constant.
Physics 151: Lecture 15, Pg 13
ACT - 3

A 0.04-kg ball is thrown from the top of a 30-m tall
building (point A) at an unknown angle above the
horizontal. As shown in the figure, the ball attains a
maximum height of 10 m above the top of the
building before striking the ground at point B. If air
resistance is negligible, what is the value of the
kinetic energy of the ball at B minus the kinetic
energy of the ball at A (KB – KA)?
Animation_1
a. 12 J
b. –12 J
c.
20 J
d. –20 J
e. 32 J
Physics 151: Lecture 15, Pg 14
Lecture 16, Example
Skateboard

What speed will skateboarder reach at bottom of the hill ?
..
Conservation of Total Energy:
m = 25 kg
v ~ 8 m/s (~16mph) !
Does NOT depend on the mass !
R=3 m
..
Physics 151: Lecture 15, Pg 15
Lecture 16, Example
Skateboard

What would be the speed if instead the skateboarder jumps
to the ground on the other side ?
..
KINEMATICS:
R=3 m
..
the same magnitude as before !
and independent of mass
Physics 151: Lecture 15, Pg 16
A Non-Conservative Force
Friction

Looking down on an air-hockey table with no air,
Path 2
Path 1
For which path does friction do more work ?
Physics 151: Lecture 15, Pg 17
A Non-Conservative Force
Path 2
Since |W2|>|W1| the puck
will be traveling slower at
the end of path 2.
Work done by a nonconservative force takes
energy out of the system.
Path 1
W1 = -mmg d1
W2 = -mmg d2
since d2 > d1,
-W2 > -W1
Physics 151: Lecture 15, Pg 18

Lecture 15, ACT 4
Work/Energy for Non-Conservative Forces
The air track is is again at an angle of 30 degrees
with respect to horizontal. The cart (with mass 1 kg)
is released 1 meter from the bottom and hits the
bumper with some speed, v1. This time the vacuum/
air generator breaks half-way through and the air
stops. The cart only bounces up half as high as
where it started. How much work did friction do on
the cart ?
30 degrees
A) 2.5 J
B) 5 J
C) 10 J
D) –2.5 J
E) –5 J
F) –10 J
Physics 151: Lecture 15, Pg 19
Generalized Work Energy Theorem:

Suppose FNET = FC + FNC (sum of conservative and nonconservative forces).

The total work done is: WTOT = WC + WNC

The Work Kinetic-Energy theorem says that: WTOT = DK.
 WTOT = WC + WNC = DK

WNC = DK - WC

But WC = -DU
So
WNC = DK + DU = DE
Physics 151: Lecture 15, Pg 20
Example - 2

A 12-kg block on a horizontal frictionless surface is
attached to a light spring (force constant = 0.80
kN/m). The block is initially at rest at its equilibrium
position when a force (magnitude P = 80 N) acting
parallel to the surface is applied to the block, as
shown. What is the speed of the block when it is 13
cm from its equilibrium position?
k
v1= 0
F
m
v2=?
0.78 m/s
x
Physics 151: Lecture 15, Pg 21
Question

1.
2.
3.
4.
5.
As an object moves from point A to point B only
two forces act on it: one force is nonconservative
and does –30 J of work, the other force is
conservative and does +50 J of work. Between A
and B,
the kinetic energy of object increases, mechanical
energy decreases.
the kinetic energy of object decreases,
mechanical energy decreases.
the kinetic energy of object decreases,
mechanical energy increases.
the kinetic energy of object increases, mechanical
energy increases.
None of the above.
Physics 151: Lecture 15, Pg 22
Question - 2

1.
2.
3.
4.
5.
As an object moves from point A to point B only
two forces act on it: one force is conservative and
does –70 J of work, the other force is
nonconservative and does +50 J of work.
Between A and B,
the kinetic energy of object increases, mechanical
energy increases.
the kinetic energy of object decreases,
mechanical energy increases.
the kinetic energy of object decreases,
mechanical energy decreases.
the kinetic energy of object increases, mechanical
energy decreases.
None of the above.
Physics 151: Lecture 15, Pg 23
ACT- 2

Objects A and B, of mass M and 2M respectively, are
each pushed a distance d straight up an inclined plane
by a force F parallel to the plane. The coefficient of
kinetic friction between each mass and the plane has
the same value . At the highest point,
1.
2.
3.
4.
KA>KB.
KA=KB.
KA<KB.
The work done by F on A is greater than the work
done on B.
5. The work done by F on A is less than the work done
on B.
Physics 151: Lecture 15, Pg 24
Recap of today’s lecture



Conservative Forces and Potential Energy
W = -DU
Conservation of mechanical energy
Physics 151: Lecture 15, Pg 25
Lecture 16, Example
Skateboard

..
Let’s now suppose that the surface is not frictionless and
the same skateboarder reach the speed of 7.0 m/s at
bottom of the hill. What was the work done by friction on
the skateboarder ?
Conservation of W + K + U = K + U
1
1
2
2
f
Total
Energy
:
m = 25 kg
Wf + 0 + mgR = 1/2mv2 + 0
Wf = 1/2mv 2 - mgR
R=3 m
..
Wf = (1/2 x25 kg x (7.0 m/s2)2 - 25 kg x 10m/s2 3 m)
Wf = 613 - 735 J = - 122 J
Total mechanical energy decreased by 122 J !
Physics 151: Lecture 15, Pg 26