Exam Review + Ch. 7: Momentum, Impulse, Center of Mass

```Physics 203 – College Physics I
Department of Physics – The Citadel
Physics 203
College Physics I
Fall 2012
S. A. Yost
Chapter 7 Part 1
+ Brief Review for Exam
Momentum and Impulse
Physics 203 – College Physics I
Department of Physics – The Citadel
Announcements
The Exam on Chapters 4 – 6 will be Thursday.
Next Tuesday: Read Ch. 7, but you can skip
sections 7.7 and 7.9.
A problem set HW07A is already open and due next
Tuesday. It covers sections 1 – 3, 8, and 9 in
Chapter 7.
A problem set HW07B on sections 4 – 6 (collisions)
will be posted soon and due next Thursday.
Physics 203 – College Physics I
Department of Physics – The Citadel
Energy Conservation
When only conservative forces act on a system,
energy is conserved.
All of the fundamental forces in the universe are
conservative.
That means that if you keep track of the total energy
in a closed system, it can never increase or
decrease – it can only change form.
Physics 203 – College Physics I
Department of Physics – The Citadel
Energy Conservation
The total energy is the sum of the kinetic energy and
the potential energy of an object.
The potential energy is the amount of work it took to
put the object in its current position. It is normally
written as U.
For example, the potential energy of a book of mass
m on top of a cabinet of height h is U = mgh.
The potential energy of a spring compressed or
stretched a distance x from equilibrium is
U = ½ kx2.
Physics 203 – College Physics I
Water Slide
Two water slides
have the
same length, but are
shaped different.
Who is going faster at
the bottom of the
slides?
A) Paul
B) Kathleen
C) No difference
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Physics 203 – College Physics I
Water Slide
Who gets to the
bottom first?
A) Paul
B) Kathleen
C) No difference
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Physics 203 – College Physics I
Department of Physics – The Citadel
Power
Power is the rate of doing work: P = W/t.
If the force F acts in the direction of motion, then
P = Fv
(instantaneous)
These are consistent because x = v t is the distance
traveled, so
P = F v = F x/t = W/t.
Horsepower:
1 hp = 746 W.
Physics 203 – College Physics I
Department of Physics – The Citadel
The Exam
Topics on Exam:
Chapter 4: Newton’s Laws
→
→
Free body diagrams, F = ma, …
Chapter 5: Circular motion, Universal gravitation.
→
→
ac = v2/R, F = ma,
Fg = Gm1m2/R,
orbits
Chapter 6: kinetic energy, work, W = DK,
potential energy, power
Physics 203 – College Physics I
Department of Physics – The Citadel
The Effect of a Force over Time
→
We say that when a force F acts for time t, a mass
acquires momentum
→
→
→
m v = m a t = F t.
If the force is changing, we can use the timeaveraged force:
→
→
m v = Favg t .
The right-hand side of the equation is called the
impulse.
Physics 203 – College Physics I
Department of Physics – The Citadel
Impulse and Momentum
→
→
The momentum can be written as p = mv.
→
→
The impulse can be written J = Favg t.
Newton’s second law implies that the net impulse
equals the change in momentum.
→
→
Dp=J
Physics 203 – College Physics I
Impulsive Forces
Momentum can be
used in any
dynamical situation,
but is especially
useful for impulsive
forces, which act
over a short time.
The impulse is the
area under the
curve,
geometrically.
Department of Physics – The Citadel
Physics 203 – College Physics I
Department of Physics – The Citadel
Impulse and Average Force
The impulse is also
equal to the average
force times the
time interval:
(same area in blue)
→
→
J(t) = Favg Dt
Physics 203 – College Physics I
Department of Physics – The Citadel
Conservation of Momentum
If there is no external force on a system of
objects, then its total momentum cannot
change, since there is no net impulse.
The total momentum of a system isolated
from external forces is conserved.
Physics 203 – College Physics I
Department of Physics – The Citadel
Center of Mass
The center of mass a set of objects is the average
position of their mass. For two objects in 1D:
x cm =
x 1 m1 + x 2 m2
m1 + m 2
x1
x2
CM
m2
m1
xcm
Physics 203 – College Physics I
Department of Physics – The Citadel
Center of Mass
In more dimensions you can use vectors to
locate the CM:
m1r1 + m2r2 + m3r3
rcm =
m1 + m2 + m3
It moves with velocity
vcm =
m1 v1 + m2v2 + m3v3
m1 + m2 + m3
m2
m1
r1
r2
CM
m2
r3
P = M vcm
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
If an external force F acts on an extended object or
collection of objects of mass M, the acceleration of the
CM is given by
F = Macm.
You can apply Newton’s 2nd Law as if it were a particle
located at the CM, as far as the collective motion is
concerned.
This says nothing about the relative motion, rotation,
etc., about the CM. That comes up in chapter 8.
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of Extended Objects
The motion of extended objects or collections of
particles is such that the CM obeys Newton’s 2nd
Law.
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
The CM of a wrench sliding on a frictionless table will move in
a straight line because there is no external force. In this
sense, the wrench may be though of as a particle located
at the CM.
cm motion
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
For example, if a hammer is thrown, its CM follows a
parabolic trajectory under the influence of gravity,
as a point object would.
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
For example, if a hammer is thrown, its CM follows a
parabolic trajectory under the influence of gravity,
as a point object would.
Physics 203 – College Physics I
Department of Physics – The Citadel
Problems for extra practice on chapter 6 follow.
Physics 203 – College Physics I
Compressed Spring
When a box is set
gently on a spring,
it compresses it a
distance d.
What would happen if
I hold it at the
uncompressed
position, and then
let go?
Department of Physics – The Citadel
Physics 203 – College Physics I
Compressed Spring
The spring will
compress – how
far?
DU = – mgh – ½ kh2
=0
h = 2mg/k
= 2d.
Department of Physics – The Citadel
Physics 203 – College Physics I
Compressed Spring
Then what happens?
Where does the box
attain its maximum
speed?
v is maximum where U
is a minimum.
This is the equilibrium
position.
Department of Physics – The Citadel
Physics 203 – College Physics I
Compressed Spring
What is the maximum speed?
Start at the top: DK +DU = 0
½ mv2 – mgd + ½ kd2 = 0
v2 = 2gd – (k/m)d2
with k = mg/d.
v2 = 2gd – gd = gd.
v = √gd
Department of Physics – The Citadel
Physics 203 – College Physics I
Department of Physics – The Citadel
Example
How long will it take 50 hp motor to pull a 120 kg sled
100 m up a hill, if the coefficient of kinetic friction is
m = 0.10, and the elevation increases by 20 m on
the way up? Assume a constant slope and speed.
Physics 203 – College Physics I
Department of Physics – The Citadel
Example
First find the work done by the motor.
Assume the net work is zero.
Wm + Wf = DU = mgh
h = 20 m
= 2.35 ×104 J.
Physics 203 – College Physics I
Department of Physics – The Citadel
Example
m = 0.10
m = 120 kg
x = 98 m
Work done by friction?
Wf = –Ff d = – m mg (cos q) d = – m mg x
4J
=
–
1.15
×10
F = mN, N = mg cos q
f
→
h = 20 m
Ff = m mg cos q
N
→
Ff
x =√ 1002 – 202 m
θ
→
mg
θ
Physics 203 – College Physics I
Department of Physics – The Citadel
Example
Wm = mgh – Wf = 2.35 ×104 J – (– 1.15 ×104 J )
= 3.50 ×104 J.
Time: t = Wm/Pm
h = 20 m
Pm = 5.0 hp (746 W/hp) = 3730 W.
t = 9.4 s.
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