Quick Concept Review and Fact
Check
Which of the following processes are due to
non-conservative forces?
A. Hockey puck slides over ice at constant speed.
B. Golf ball rolls to a stop over rough grass.
C. Car moves at constant speed with constant power from engine.
1.
2.
3.
4.
5.
A only
B only
C only
A and B
B and C
Name the following terms in the correct order.
1 2
Kx
2
1
Mv 2
2
1. Gravitational potential, Kinetic
energy, Spring Potential
2. Spring potential, Kinetic
energy, gravitational potential.
3. Kinetic energy, gravitational
potential, Spring potential.
MgH
Chapter 8.4-8.5
Work, Force and Energy redux.
The most important messages are these:
Mechanical energy is given by:
E=K+U
Energy changes only if there are non-conservative forces:
DE = Ef – Ei = Wnc
This means that Mechanical Energy is CONSERVED if Wnc = 0.
SUMMARY RELATIONSHIPS:
Wtotal = Wc + Wnc = DK
Wc = - DU
Wnc = DE
This is a type of “book-keeping” for energy.
Starting with the definition of mechanical energy…
E=K+U
Then the changes in the energy terms are also related….
DE = DK + DU
Each energy change is associated with a type of work….
DE = DK + DU
Wnc = Wtot - Wc
And if we re-arrange the terms, we get…
Wtot = Wc + Wnc
What does this mean, “in words”?
DE = DK + DU
Wnc = Wtot - Wc
The non-conservative work, Wnc, goes into heat (or sound, etc.),
and is “lost” from the mechanical system. Wnc is NEGATIVE!
(almost always)
Example is friction. The force of friction acts opposite to the direction
of motion, so W = F x D is NEGATIVE.
In words….. 1. The mechanical energy is CHANGED by the amount of
non-conservative work done by the forces in the system.
(Heat energy going into or out of the system)
2. The Kinetic Energy is INCREASED by the amount of total
work done by forces in the system (conservative plus nonconservative).
3. The potential energy changes OPPOSITE to the work
done by conservative forces in the system. (Potential
stores energy of conservative forces)
Conservative Work: Atwood’s machine
Q: An “Atwood’s Machine” is shown in Fig. 1 at the left. What is the
speed of the two masses just before mass #2 hits the ground?
Potential
DECREASES
E INIT= 0 (my choice of potential)
1
1
m1v 2  m2 v 2  m1 gh  m2 gh
2
2
0
EFINAL 
DE = 0 (conservative forces)
Potential
INCREASES
So, Efinal = Einit
1
m1  m2 v 2  m2  m1 gh
2
CHECK: Suppose m1 was equal to zero? What if masses are equal?
How much does spring compress?
K
DX
V
M
V=0
Solve using energy conservation:
What is initial mechanical energy? (Kinetic and potential)
What is final mechanical energy? (Kinetic and potential)
Set them equal.
Solve in class (next slide).
How much does the spring compress?
K
DX
M
V
V=0
A
MV
L
K
1. A.
2. B.
3. C.
B
Mg
L
K
C
L V
M
K
A mass M moving at speed V
hits an initially relaxed spring of
constant K. What is the
maximum compression of the
spring, DX?
Non-conservative Work
V0
V1
L
M
M
Rough spot, coefficient
of friction m.
A block of mass M is moving initially with speed V0. It
passes over a rough patch of table of length L. Given the
mass M, the initial speed V0, and the coefficient of friction,
find the final speed of the block.
1. Draw the picture. 2. What is given? 3. What are the relationships?
Non-conservative work: WNC  F  L   mMgL
Work-energy:
WNC  DE
DE = E1 – E0
Set up the problem
V0
V1
L
M
M
Rough spot, coefficient
of friction m.
A block of mass M is moving initially with speed V0. It
passes over a rough patch of table of length L. Given the
mass M, the initial speed V0, and the coefficient of
friction, find the final speed of the block.
1.
2.
3.
4.
5.
What is the initial total energy?
What is the final total energy?
What is the change in energy?
What is the non-conservative work (Force x distance)?
Use WNC = DE.
Solve in class (next slide).
What is the final speed?
V1
L
V0
M
M
Rough spot, coefficient
of friction m.
A.
V1  V02  2mgL
B.
V1  V0  2mgL
C.
V1  V02  2mgL
Hard by force, easy by energy.
M
Ball of mass M slides without
friction down the ramp, and then
slides WITH friction (m) on the
grass. How far does it slide, L?
H
L
1.
2.
3.
4.
What is the initial energy?
What is the final energy?
What is the work done? (Hint: non-conservative)
Use Wnc = DE.
Ball rolling down slide.
Ball of mass M slides without
friction down the ramp, and then
slides WITH friction (m) on the
grass. How far does it slide, L?
H
g
A.
L
B.
L
C.
L  mH
m
H
m
L
Combo problem.
Ball slides down ramp,
and then falls by free-fall
a height Y. What is range
X?
H
V
Y
X
Conservation of energy:
1
Mv 2
2
v  2 gH
MgH 
X  2 gH
Kinematics:
2Y
 2 YH
g
X v
2Y
g