Physics 151: Lecture 16
Today’s Agenda

Today’s Topics:
Conservation of mechanical energy
Nonconservative forces and loss of energy from
a system
Other relationships for potential energy
Physics 151: Lecture 16, Pg 1
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
E = K + U
= W + U
= W + (-W) = 0
using K = W
using U = -W
E = K + U is constant !!!
Both K and U can change, but E = K + U remains constant.
Physics 151: Lecture 16, Pg 2
See text: Example 8.3
Example: The simple pendulum.

Suppose we release a mass m from rest a distance h1
above its lowest possible point.
 What is the maximum speed of the mass and where
does this happen ?
 To what height h2 does it rise on the other side ?
m
h1
h2
v
Physics 151: Lecture 16, Pg 3
See text: Example 8.3
Example: The simple pendulum.


Energy is conserved since gravity is a conservative force
(E = K + U is constant)
Choose y = 0 at the bottom of the swing,
and U = 0 at y = 0 (arbitrary choice).
E = 1/2mv2 + mgy.
y
y=0
h1
h2
v
Physics 151: Lecture 16, Pg 4
Lecture 16, ACT 1
The Roller Coaster

I have built a Roller Coaster . A motor tugs the
cars to the top and then they are let go and are in
the hands of gravity. To make the following loop,
how high do I have to let the release the car ??
Car has mass m
h?
R
A) 2R
B) 5R
C) 5/2 R
D) none of the above
Physics 151: Lecture 16, Pg 5
Another Problem Involving a Spring
A large spring is used to stop the cars after they come down the
last hill of a roller coaster. The cars start at rest at the top of the
hill and are caught by a mechanism at the instant their velocities
at the bottom are zero. Compare the compression of the spring,
xA, for a fully loaded car with that, xB, for a lightly loaded car
when mA= 2mB.
a. xA = 1/2 xB.
b. xA = xB.
c. xA = (2)1/2 xB.
d. xA = 2 xB.
e. xA = 4 xB.
Physics 151: Lecture 16, Pg 6
See text: 8.5
Non-conservative Forces:

If the work done does not depend on the path taken, the
force involved is said to be conservative.

If the work done does depend on the path taken, the force
involved is said to be non-conservative.

An example of a non-conservative force is friction:

Pushing a box across the floor, the amount of work that is
done by friction depends on the path taken.
 Work done is proportional to the length of the path !
Physics 151: Lecture 16, Pg 7
Generalized Work Energy Theorem:
WNC = K + U = E
Physics 151: Lecture 16, Pg 8
Lecture 16, ACT 2
Stones and Friction

I throw a stone into the air. While in flight it feels
the force of gravity and the frictional force of the
air resistance. The time the stone takes to reach
the top of its flight path (i.e. go up) is,
A) larger than
B) equal to
C) less than
The time it takes to return from the top (i.e. go
down).
Physics 151: Lecture 16, Pg 9
Lecture 16, ACT 2
Solution

We know that WNC = K + U = E

Thus as time progresses the amount of energy in
the earth-rock system is continually decreasing

If I consider the way up versus the way down at a
given height,
then Eup = Kup + Uup > Edown = Kdown + Udown

But, at a given height U is always the same.
So, Kup > Kdown

v at a given height is always less on the way
down
The answer is (C) it takes less time to go up
Physics 151: Lecture 16, Pg 10
Same problem but with numbers:
• A 2.0-kg mass is projected vertically upward from ground
level with an initial speed of 30 m/s. The mass rises to a
maximum height of 35 m above ground level. How much
work is done on the mass by air resistance between the
point of projection and the point of maximum height?
a. –0.21 kJ
b. +0.21 kJ
c. –0.40 kJ
d. +0.69 kJ
e. –0.69 kJ
Physics 151: Lecture 16, Pg 11
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 16, Pg 12
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 16, Pg 13
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.
5.
KA>KB.
KA=KB.
KA<KB.
The work done by F on A is greater than the work done on B.
The work done by F on A is less than the work done on B.
Physics 151: Lecture 16, Pg 14
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 16, Pg 15
See Text: section 8.6
Conservative Forces and Potential Energy

We have defined potential energy for conservative forces
U = -W

But we also now that
W = Fxx

Combining these two,
U = - Fxx

Letting small quantities go to infinitesimals,
dU = - Fxdx

Or,
Fx = -dU/dx
Physics 151: Lecture 16, Pg 16
See Text: section 8.6
Examples of the U - F relationship

Remember the spring,
U = (1/2)kx2

Do the derivative
Fx = - d ( (1/2)kx2) / dx
Fx = - 2 (1/2) kx
Fx = -kx
Physics 151: Lecture 16, Pg 17
See Text: section 8.6
Examples of the U - F relationship

Remember gravity,
Fy = mg

Do the integral
U = - F dy
U = - (-mg) dy
U = mgy + C
U = U2 – U1 = (mgy2 + C) – (mgy1 + C) = mgy
Physics 151: Lecture 16, Pg 18
Recap of today’s lecture



Conservation of Mechanical Energy
Non-conservative forces and loss of energy from
a system
Other relationships for potential energy
Physics 151: Lecture 16, Pg 19