Download Chapter_7

Chapter 7:
Work and Kinetic Energy-1
Reading assignment: Chapter 7.6-7.9
Homework: due Monday, Sept 28, 2009
Chapter 7: 5AE's, 5AF's, Q7, 2, 9, 18, 22
• The concept of energy (and the conservation of energy)
is one of the most important topics in physics.
• Work
• Dot products
• Energy approach is simpler than Newton’s second law.
1. A woman holds a bowling ball in a fixed position.
The work she does on the ball
___ 1. depends on the weight of the ball.
___ 2. cannot be calculated without more information.
___ 3. is equal to zero.
2. A man pushes a very heavy load across a horizontal
floor. The work done by gravity on the load
___ 1. depends on the weight of the load.
___ 2. cannot be calculated without more information.
___ 3. is equal to zero.
3. When net positive work is done on a particle, its kinetic
energy
a. increases.
b. decreases.
c. remains the same.
d. need more information about the way the work was done
4. In a collision between two billiard balls,
a. energy is not conserved if the collision is perfectly elastic.
b. momentum is not conserved if the collision is inelastic.
c. not covered in the reading assignment
Work
(as defined by a physicist)
Definition:
The work done on an object by an external force is
- the product of the component of the force in the direction
of the displacement and the magnitude of the displacement.
W  F  d  cos
How much work is done
when just holding up an
object?
W  F  d  cos 
W 0
How much work is done
when the displacement is
perpendicular to the
force?
W  F  d  cos 
W 0
F = 10 N
What is the work done by
 = 60°
- the gravitational force
- the normal force
- the force F
when the block is displaced
along the horizontal.
The total work is:
- the sum of the work done by all forces


- or: Wtotal  Fnet  d net
d = 10m
What is the work done when lifting?
- By the gravitational force?
- By the applied force?
W  F  d  cos 
W  F  d  1372J
Strongest man lifting 140 kg boulder by 1 m.
Sign convention:
W is positive:
If F and d are parallel
If energy is transferred into the system
W is positive:
If F and d are antiparallel
If energy is transferred out of the system
Work is a scalar quantitiy. (not a vector)
Work has units of
newton·meter (N·m) = the joule (J)
Black board example 7.5
A man loads a refrigerator onto a truck using a ramp. Ignore
friction.
He claims he would be doing less work if the length of the ramp
would be longer. Is this true?
Black board example 7.1
A donkey is pulling a cart
with a force of magnitude
F = 500 N at an angle of
30º with the horizontal.
Calculate the work done by
the donkey as the cart is
pulled for one mile (1648
m).
500 N
Definition of dot product and work
Work is the scalar product (or dot product)
of the force F and the displacement d.
 
W  F  d  F  d  cos
F and d are vectors
W is a scalar quantity
Scalar product between
vector A and B
Definition:
 
A  B  AB  cos
Scalar product is commutative:
   
A B  B  A
Distributive law of multiplication:
  
   
A  (B  C)  A  B  A  C
Scalar Product using unit vectors:
We have the vectors A and B:




A  Ax i  Ay j  Az k
Then:
 
A  B  Ax Bx  Ay By  Az Bz
 
A  A  Ax Ax  Ay Ay  Az Az  A2




B  Bx i  By j  Bz k
Black board example 7.2
A particle moving in the x-y plane undergoes a
displacement d = (2.0i + 3.0j) m at a constant force
F = (5.0i + 2.0j) N acts on the particle. Calculate
(a) The magnitude of the displacement and the force.
(b) The work done by F.
(c) The angle between F and d.
1. The gravitational potential energy of a particle at a height z above
Earth’s surface
___ 1. depends on the height z.
___ 2. depends on the path taken to bring the particle to z.
___ 3. both 1 and 2.
___ 4. is not covered in the reading assignment.
2. Which of the following is not a conservative force?
___ 1. the force exerted by a spring on a particle in one dimension
___ 2. the force of friction
___ 3. the force of gravity
___ 4. not covered in the reading assignment
3. Which of the following was not discussed in the reading assignment?
___ 1. conservation of non-conservative forces
___ 2. block and tackle
___ 3. work
___ 4. all of the above were discussed
Black board example 7.2
A particle moving in the x-y plane undergoes a
displacement d = (2.0i + 3.0j) m at a constant force
F = (5.0i + 2.0j) N acts on the particle. Calculate
(a) The magnitude of the displacement and the force.
(b) The work done by F.
(c) The angle between F and d.
What if the force varies? We have to integrate the force along x
xf
Work done by a varying force:
W   F  x dx
xi
Thus, the work is equal to the area under the F(x) vs. x curve.
Black board example 7.3
A force acting on a particle
varies as shown in the
Figure.
What is the total work done on the particle as it is moved from x =
0 to x = 8 m?
Hint: It is the area under the curve.
Consider a spring
Hooke’s law:
(Force required to stretch or
compress a spring by x):
Fs  k  x
k is the spring constant of a spring.
Stiff springs have a large k value.
Work done by a spring
xi
1
2
2
W  k ( xi  x f )
2
xf
A spring-loaded toy dart gun is used to shoot
a dart straight up in the air, and the dart
reaches a maximum height of 24 m.The same
dart is shot straight up a second time from
the same gun, but this time the spring is compressed
only half as far before firing. How
far up does the dart go this time, neglecting
friction and assuming an ideal spring?
1. 96 m
2. 48 m
3. 24 m
4. 12 m
5. 6 m
6. 3 m
7. impossible to determine
Black board example 7.3
A 0.500 kg mass is hung
from a spring extending
the spring by a distance
x = 0.2 m
(a) What is the spring constant of the spring?
(b) How much work was done on the mass by the gravitational force
(c) How much work was done on the mass by the spring force?
The kinetic energy of a particle is:
1
2
K  mv
2
A bullet of mass m = 0.020 kg moves at 500 m/s.
A truck of mass m = 1000 kg moves at 5 m/s
Which has more kinetic energy?
Work due to friction
If friction is involved in moving objects, work has to be
done against the kinetic frictional force.
This work is:
Wf  fk  d
A cart on an air track is moving at 0.5 m/s
when the air is suddenly turned off. The cart
comes to rest after traveling 1m. The experiment
is repeated, but now the cart is moving
at 1 m/s when the air is turned off. How far
does the cart travel before coming to rest?
1. 1 m
2. 2 m
3. 3 m
4. 4 m
5. 5 m
6. impossible to determine
Black board example 7.4
Angus is pulling a 10,000 kg truck with all his
might (2000N) on a frictionless surface
for 10.0 m.
(a) How much work is the man doing?
(b) What is the speed of the truck after 10 m.
(c) What is the speed of the truck after 10 m
if there is friction?
(friction coefficient: 0.0153)
Power
Power is the rate at which work is done:
dW
P
dt
Average power:
(work done per time
interval Dt)
W
P
Dt
The power can also be expressed as:

dW  ds  
P
F
 F v
dt
dt
(Dot product)
The units of power are joule/sec (J/s) = watt (W)
Black board example 7.7
An elevator having a total mass of
3000 kg moves upward against
the gravitational force at a
constant speed of 9.13 m/s.
(a) What is the power delivered by
the motor?
Chapter 8: Potential Energy and
Conservation of Energy part 1 (finish Chp7)
Reading assignment: Chapter 8.5-8-7
Homework : due Monday, October 5, 2009
Problems:
Chapter 8 33, 36, 37, 58, 65
Bonus: 48, 64, 37
• One form of energy can be converted
into another form of energy.
• Conservative and non-conservative
forces
• CONSERVATION OF ENERGY
1. Suppose you know the potential energy function corresponding to a
force. Is it always possible to calculate the force?
___ 1. yes
___ 2. only if the force is nonconservative
___ 3. not covered in the reading assignment
2. The potential energy of a spring is
___ 1. proportional to the amount the spring is stretched.
___ 2. proportional to the square of the amount the spring is stretched.
___ 3. not yet covered in any reading assignment.
3. A car slows down as a result of air friction. Which is true?
___ 1. The car’s kinetic energy decreases.
___ 2. Heat is generated.
___ 3. The energy of the car/road/air system is constant.
___ 4. all of the above
___ 5. none of the above
Potential energy U:
- Can be thought of as stored energy that can either
do work or be converted to kinetic energy.
- When work gets done on an object, its potential
and/or kinetic energy increases.
- There are different types of potential energy:
1. Gravitational energy
2. Elastic potential energy (energy in an stretched spring)
3. Others (magnetic, electric, chemical, …)
Conservative and non-conservative forces
Conservative forces:
Work is independent of the path taken.
Work depends only on the final and initial point.
Work done is zero if the path is a closed loop (same
beginning and ending points.)
We can always associate a potential energy with
conservative forces.
We can only associate a potential energy with
conservative forces.
Work done by a conservative force: Wc = Ui – Uf = - DU
Examples of conservative forces:
_____________________________________________
Conservative forces and potential energy
xf
Wc   F  x dx   DU
xi
dU   Fx dx
Thus,
dU
Fx  
dx
The work done by a conservative
force equals the negative of the
change in potential energy
associated with that force.
Any conservative force acting on an
object within a system equals the
negative derivative of the potential
energy of the system with respect to x.
Conservative and non-conservative forces
Non-conservative forces:
A force is non-conservative if it causes a change in mechanical
energy; mechanical energy is the sum of kinetic and potential
energy.
Example: Frictional force.
- This energy cannot be converted back into other forms of
energy (irreversible).
- Work does depend on path.
Sliding a book on a table
Gravitational
potential energy:
Ug  m g  y
- Potential energy only depends on y (height) and not on x (lateral distance)
DU g  U f  U i  mg ( y f  yi )
Black board example 8.1
You are 1.80 m tall.
A 0.1 kg apple, which is hanging 1 m
above your head, drops on you.
What is the difference in
gravitational potential energy
when it hangs and when it hits
you?
(a) How much gravitational potential
energy does it loose?
1m
Work done by/on a spring:
1
2
2
W  k ( xi  x f )
2
xi
xf
Elastic potential energy stored in a spring:
1 2
U  kx
2
The spring is stretched or
compresses from its
equilibrium position by x
Review Important energy formulas:
Work:
 
W  F d
 F  d  cos 
xf
W   F  x dx
 Fx  d x  Fy  d y  Fz  d z
xi
Forms of energy:
Kinetic energy :
1
K  m  v2
2
Gravitatio nal potential energy :
Ug  m g h
1 2
Elastic potential energy : U e  kx
2
Demo example (conversion of energy):
(ballistic pendulum)
Conversion of:
Elastic potential energy
into kinetic energy
into gravitational potential energy
Black board example 8.2
A mass m is bobbing up and down on a
spring.
Describe the various forms of energy
of this system.
(a) At the highest point
(b) At the point where the kinetic
energy is highest
(c) At the lowest point
Black board example 8.3
Three balls are thrown from the
top of a building, all with
the same initial speed.
The first is thrown horizontally,
the second with some angle
above the horizontal and the
third with some angle below
the horizontal.
(a) Describe the motion of the balls.
(b) Rank the speed of the balls as they hit the ground.
Black board example 8.5
Nose crusher?
A bowling ball of mass m
is suspended from the
ceiling by a cord of
length L. The ball is
released from rest
when the cord makes
an angle A with the
vertical.
(a) Find the speed of the ball at the lowest point B.
(b) Assume a cord length L = 5m and an angle A = 20°.
(c) The ball swings back. Will it crush the operator’s nose?
Reading potential energy curves
Remember:
dU
Fx  
dx
E  K U
Black board example 8.6
During a rock slide, a 520 kg rock slides from rest down a hillside that is
500 m long and 300 m high. The coefficient of friction between the
rock and the hillside is 0.25.
(a) What is the gravitational potential energy of the rock before the
slide?
(b) How much energy is transferred into thermal energy during the slide?
(c) What is the kinetic energy of the rock as it reaches the bottom of the
hill?
Chapter 8: Potential Energy and
Conservation of Energy part 2
Reading assignment: Chapter 9
Homework : (due Wednesday, Oct. 7, 2009):
Problems:
10AE's, Q2, 3, 6, 19, 20
• One form of energy can be converted
into another form of energy.
• Conservative and non-conservative
forces
• CONSERVATION OF ENERGY
(a) Can an object-Earth system have kinetic energy and not gravitational
potential energy?
Yes
No
(b) Can it have gravitational potential energy and not kinetic energy?
Yes
No
(c) Can it have both types of energy at the same moment?
Yes
No
(d) Can it have neither?
Yes
No
1) A ball of clay falls freely to the hard floor. It does not bounce noticeably, but very
quickly comes to rest. What then has happened to the energy the ball had while it was
falling?
a) Most of it went into sound.
b) It has been transformed back into potential energy.
c) It is in the ball and floor as energy of invisible molecular motion.
d) It has been used up in producing the downward motion.
e) It has been transferred into the ball by heat.
2) You hold a slingshot at arm's length, pull the light elastic band back to your chin, and
release it to launch a pebble horizontally with speed 100 cm/s. With the same
procedure, you fire a bean with speed 500 cm/s. What is the ratio of the mass of the
bean to the mass of the pebble (bean/pebble)?
a) 1/5
b) 1/√5
c) 1
d) √5
e) 5
Conservation of mechanical energy
If we deal only with conservative forces and
If we deal with an isolated system (no energy added or removed):
The total mechanical energy of a system remains constant!!!!
E  K U
E… total energy
K… Kinetic
energy
U… potential
The final and initial energy of a system remain the same:
Ei = Ef
energy
Thus:
E  Ki  U i  K f  U f
Work due to friction
If friction is involved in moving objects, work has to be
done against the kinetic frictional force.
This work is:
Wf  fk  d
Work done by non-conservative forces
1. Work done by an applied force.
(System is not isolated)
An applied force can transfer energy into or out of the system.
Example. Applying a force to an object and lifting increases the
energy of the object.
 
W  F  d  F  d  cos
Work done by non-conservative forces
2. Situations involving kinetic friction.
(Friction is not a conservative force).
Kinetic friction is an example of a non-conservative force.
If an object moves over a surface through a distance d, and it
experiences a kinetic frictional force of fk it is loosing kinetic
energy
DK friction  DW friction   f k  d
Thus, the mechanical energy (E = U + K) of the
system is reduced by this amount.
Black board example 8.4
HW 12
A frictionless roller coaster with an initial speed of v0 = 10.00 m/s,
at the initial height h = 100.0 m, has a mass m = 1000.0 kg
(a) What is the speed at point A?
(b) What is the speed at point B
(c) How high will it move up on the last hill?
3) A pile driver is a device used to drive posts into the Earth by
repeatedly dropping a heavy object on them. Assume the
object is dropped from the same height each time. By what
factor does the energy of the pile driver-Earth system change
when the mass of the object being dropped is tripled?
a) 1/9
b) 1/3
c) 1: the energy is the same
d) 3
e) 9
A curving children's slide is installed next to a backyard swimming pool. Two children
climb to a platform at the top of the slide. The smaller child hops off to jump straight
down into the pool and the larger child releases herself at the top of the frictionless
slide.
(4) Upon reaching the water, how does the kinetic energy of the larger child compare to
that of the smaller child?
a) greater than
b) equal to
c) less than
(5) Upon reaching the water, how does the speed of the larger child compare to that of
the smaller child?
a) equal to
b) less than
c) greater than
(6) During the motions from the platform to the water, how does the acceleration of the
larger child compare to that of the smaller child?
a) equal to
b) less than
c) greater than
Black board example 8.7
A 3.5 kg block is accelerated by a compressed spring whose spring
constant is 640 N/m. After leaving the spring at the spring’s relaxed
length, the block travels over a horizontal surface with a frictional
coefficient mk = 0.25 for a distance of 7.8 m.
(a) What is the increase in the thermal energy of the block-floor system?
(b) What is the maximum kinetic energy of the block?
(c) Through what distance was the spring compressed initially (before
the block moved)?
Suppose you drop a 1-kg rock from a height of 5
m above the ground. When it hits, how much
force does the rock exert on the ground?
1. 0.2 N
2. 5 N
3. 50 N
4. 100 N
5. impossible to determine
Chapter 9: Linear Momentum &
Collisions
Reading assignment: Chapter 9.5-9.7
Homework : (due Saturday, Oct. 10, 2009):
Problems:
Chapter 9: 5AF's, Q11, 28, 32, 40, 68, 69
• Center of mass
• Momentum


p  mv
• Momentum is conserved
1. The impulse delivered to a body by a force is
___ 1. defined only for interactions of short duration.
___ 2. equal to the change in momentum of the body.
___ 3. equal to the area under an F vs. x graph.
___ 4. defined only for elastic collisions.
2. In an elastic collision
___ 1. energy is conserved.
___ 2. momentum is conserved.
___ 3. the magnitude of the relative velocity is conserved.
___ 4. all of the above
3. In an inelastic collision
___ 1. both energy and momentum are conserved.
___ 2. energy is conserved.
___ 3. momentum is conserved.
___ 4. neither is conserved.
4. In two-dimensional elastic collisions, the conservation laws
___ 1. allow us to determine the final motion.
___ 2. place restrictions on possible final motions.
___ 3. do not allow us to say anything about the final motion.
___ 4. are not covered in the reading assignment.
Suppose you drop a 1-kg rock from a height of 5
m above the ground. When it hits, how much
force does the rock exert on the ground?
1. 0.2 N
2. 5 N
3. 50 N
4. 100 N
5. impossible to determine
Center of mass
Center of mass for many particles:

rCM 

 mi ri
i
M
Black board example 9.1
Where is the center of mass of the arrangement of particles below.
(m3 = 2 kg and m1 = m2 = 1 kg)?
A method for finding the center of mass of any object.
- Hang object from two
or more points.
- Draw extension of
suspension line.
- Center of mass is at
intercept of these lines.
Center of mass of a solid body
(uniform density)
xCM
yCM
zCM
1
  xdV
V
1
  ydV
V
1
  zdV
V
Black board example 9.2
A uniform square plate 6 m
on a side has had a square
piece 2 m on a side cut out of
it. The center of that piece is
at x = 2 m, y = 0. The center
of the square plate is at x = y
= 0. Find the coordinates of
the center of mass of the
remaining piece.
Motion of a System of Particles.
Newton’s second law for a System of Particles
The center of mass of a system of particles (combined mass M)
moves like one equivalent particle of mass M would move under
the influence of an external force.


Fnet  MaCM
Fnet , x  MaCM , x
Fnet , y  MaCM , y
Fnet , z  MaCM , z
A rocket is shot up in the air and explodes.
Describe the motion of the center of mass before and after
the explosion.
Linear Momentum
The linear momentum of a particle of mass m and velocity
v is defined as


p  mv
The linear momentum is a vector quantity.
It’s direction is along v.
The components of the momentum of a particle:
px  m  vx
py  m  vy
pz  m  vz




dp d (m  v )
From Newton’s second law: Fnet 

 ma
dt
dt
The time rate of change in linear momentum is equal to the net
forces acting on the particle.
This is also true for a system of particles:


P  M  vCM
Total momentum = Total mass ·velocity of center of mass
And: Net external force = rate of change in
momentum of the center of mass


dP
Fnet 
dt
Conservation of linear momentum
Thus:
If no external force is acting on a particle, it’s momentum is
conserved.
This is also true for a system of particles:
If no external forces interact with a system of particles the total
momentum of the system remains constant.

  
P   p  p1  p2      constant
 
or : Pi  Pf




p1i  p2i      p1 f  p2 f    
Suppose you are on a cart, initially at rest on
a track with very little friction. You throw
balls at a partition that is rigidly mounted on
the cart. If the balls bounce straight back as
shown in the figure, is the cart put in motion?
1. Yes, it moves to the right.
2. Yes, it moves to the left.
3. No, it remains in place.
Black board example 9.3
You (100kg) and your skinny friend
(50.0 kg) stand face-to-face on a
frictionless, frozen pond. You push
off each other. You move backwards
with a speed of 5.00 m/s.
(a) What is the total momentum of the
you-and-your-friend system?
(b) What is your momentum after you
pushed off?
(c) What is your friends speed after you
pushed off?
Chapter 9: Linear Momentum and
Collisions
Reading assignment: Chapter 10.1-10.5
Homework #16 :
Problems:
(due Monday, Oct. 10, 2005):
Q1, Q14, 9, 14, 21, 28
• Momentum


p  mv
• Momentum is conserved – even in collisions with energy
loss due to friction/deformation.
• Impulse
1) If two objects collide and one is initially at
rest, is it possible for both to be at rest after the
collision?
A) yes
B) no
2) Is it possible for one to be at rest after the
collision?
A) yes
B) no
Elastic and inelastic collisions in one dimension
Momentum is conserved in any collision, elastic and
inelastic.
Kinetic Energy is only conserved in elastic collisions.
Perfectly inelastic collision: After colliding, particles stick
together. There is a loss of kinetic energy (deformation).
Inelastic collisions: Particles bounce off each with some loss
of kinetic energy.
Perfectly elastic collision: Particles bounce off each other
without loss of kinetic energy.
Perfectly inelastic collision of two particles
(Particles stick together)
 
pi  p f



m1v1i  m2v2i  (m1  m2 )v f
Notice that p and v are
vectors and, thus have a
direction (+/-)
K i  Eloss  K f
1
1
1
2
2
2
m1v1i  m2v2i  Eloss  (m1  m2 )v f
2
2
2
There is a loss
in kinetic
energy, Eloss
Elastic collision of two particles
(Particles bounce off each other without loss of energy.
Momentum is conserved:




m1v1i  m2 v2i  m1v1 f  m2v2 f
Energy is conserved:
1
1
1
1
2
2
2
2
m1v1i  m2 v2i  m1v1 f  m2 v2 f
2
2
2
2
For elastic collisions in one dimension:
Suppose we know the initial masses and velocities.
Then:
 m1  m2 
 2m2 
v1i  
v2i
v1 f  
 m1  m2 
 m1  m2 
(9.21)
 2m1 
 m2  m1 
v1i  
v2i
 
 m1  m2 
 m1  m2 
(9.22)
v2 f
Black board example 9.2
Two carts collide elastically on a frictionless track. The first
cart (m1 = 1kg) has a velocity in the positive x-direction of
2 m/s; the other cart (m = 0.5 kg) has velocity in the
negative x-direction of 5 m/s.
(a) Find the speed of both carts after the collision.
(b) What is the speed if the collision is inelastic?
(c) How much energy is lost in the inelastic collision?
Two-dimensional collisions (Two particles)
Conservation of momentum:




m1v1i  m2 v2i  m1v1 f  m2v2 f
Split into components:
m1v1ix  m2v2ix  m1v1 fx  m2v2 fx
m1v1iy  m2v2iy  m1v1 fy  m2v2 fy
If the collision is elastic, we can also use conservation of energy.
Black board example 9.3
Accident investigation. Two
automobiles of equal mass approach
an intersection. One vehicle is
traveling towards the east with 29 mi/h
13.0 m/s
(13.0 m/s) and the other is traveling
north with unknown speed. The
vehicles collide in the intersection and
stick together, leaving skid marks at an
angle of 55º north of east. The second
driver claims he was driving below the
speed limit of 35 mi/h (15.6 m/s).
??? m/s
Is he telling the truth?
What is the speed of the “combined vehicles” right after the collision?
How long are the skid marks (mk = 0.5)
If ball 1 in the arrangement shown here is
pulled back and then let go, ball 5 bounces
forward. If balls 1 and 2 are pulled back and
released, balls 4 and 5 bounce forward, and
so on. The number of balls bouncing on each
side is equal because
1. of conservation of momentum.
2. the collisions are all elastic.
3. neither of the above
Impulse (change in momentum)

 

A change in momentum is called “impulse”: J  Dp  p f  pi
During a collision, a force F acts on
an object, thus causing a change in
momentum of the object:
For a constant (average) force:
tf


Dp  J   F (t )dt
ti
 
Dp  J  Favg  Dt
Think of hitting a soccer ball: A force F acting over a time Dt
causes a change Dp in the momentum (velocity) of the ball.
Black board example 10.1
A soccer player hits a ball
(mass m = 440 g) coming
at him with a velocity of
20 m/s. After it was hit,
the ball travels in the
opposite direction with a
velocity of 30 m/s.
(a) What impulse acts on the
ball while it is in contact
with the foot?
(b) The impact time is 0.1s.
What is the force acting
on the ball?
Chapter 9: Linear Momentum and
Collisions
Reading assignment: Chapter 10.1-10.5
Homework #16 :
Problems:
(due Monday, Oct. 10, 2005):
Q1, Q14, 9, 14, 21, 28
• Momentum


p  mv
• Momentum is conserved – even in collisions with energy
loss due to friction/deformation.
• Impulse
1) If two objects collide and one is initially at
rest, is it possible for both to be at rest after the
collision?
A) yes
B) no
2) Is it possible for one to be at rest after the
collision?
A) yes
B) no
Elastic and inelastic collisions in one dimension
Momentum is conserved in any collision, elastic and
inelastic.
Kinetic Energy is only conserved in elastic collisions.
Perfectly inelastic collision: After colliding, particles stick
together. There is a loss of kinetic energy (deformation).
Inelastic collisions: Particles bounce off each with some loss
of kinetic energy.
Perfectly elastic collision: Particles bounce off each other
without loss of kinetic energy.
Perfectly inelastic collision of two particles
(Particles stick together)
 
pi  p f



m1v1i  m2v2i  (m1  m2 )v f
Notice that p and v are
vectors and, thus have a
direction (+/-)
K i  Eloss  K f
1
1
1
2
2
2
m1v1i  m2v2i  Eloss  (m1  m2 )v f
2
2
2
There is a loss
in kinetic
energy, Eloss
You are given two carts, A and B. They look
identical, and you are told that they are made of the
same material. You place A at rest on an air track
and give B a constant velocity directed to the right
so that it collides with A. After the collision, both
carts move to the right, the velocity of B being
smaller than what it was before the collision. What
do you conclude?
1. Cart A is hollow.
2. The two carts are identical.
3. Cart B is hollow.
4. need more information
A car accelerates from rest. In doing so the
car gains a certain amount of momentum
and Earth gains
1. more momentum.
2. the same amount of momentum.
3. less momentum.
4. The answer depends on the interaction
between the two.
A car accelerates from rest. It gains a certain
amount of kinetic energy and Earth
1. gains more kinetic energy.
2. gains the same amount of kinetic energy.
3. gains less kinetic energy.
4. loses kinetic energy as the car gains it.
Suppose the entire population of the world gathers in
one spot and, at the sounding of a prearranged signal,
everyone jumps up. While all the people are in the air,
does Earth gain momentum in the opposite direction?
1. No; the inertial mass of Earth is so large that the
planet’s change in motion is zero.
2. Yes; because of its much larger inertial mass,
however, the change in momentum of Earth is much
less than that of all the jumping people.
3. Yes; Earth recoils, like a rifle firing a bullet, with a
change in momentum equal to and opposite that of the
people.
4. It depends.
Suppose the entire population of the world
gathers in one spot and, at the sound of a
prearranged signal, everyone jumps up. About a
second later,5 billion people land back on the
ground. After the people have landed, Earth’s
momentum is
1. the same as what it was before the people
jumped.
2. different from what it was before the people
jumped.
Suppose rain falls vertically into an open
cart rolling along a straight horizontal track
with negligible friction. As a result of the
accumulating water, the speed of the cart
1. increases.
2. does not change.
3. decreases.
Suppose rain falls vertically into an open
cart rolling along a straight horizontal track
with negligible friction. As a result of the
accumulating water, the kinetic energy of
the cart
1. increases.
2. does not change.
3. decreases.
Consider these situations:
(i) a ball moving at speed v is brought to rest;
(ii) the same ball is projected from rest so that it
moves at speed v;
(iii) the same ball moving at speed v is brought to rest
and then projected backward to its original speed.
In which case(s) does the ball undergo the largest
change in momentum?
1. (i)
2. (i) and (ii)
3. (ii)
4. (ii) and (iii)
5. (iii)
Consider two carts, of masses m and 2m, at
rest on an air track. If you push first one cart
for 3 s and then the other for the same length
of time, exerting equal force on each, the
momentum of the light cart is
1. four times
2. twice
3. equal to
4. one-half
5. one-quarter
the momentum of the heavy cart.
Consider two carts, of masses m and 2m, at
rest on an air track. If you push first one cart
for 3 s and then the other for the same length
of time, exerting equal force on each, the kinetic
energy of the light cart is
1. larger than
2. equal to
3. smaller than
the kinetic energy of the heavy car.
Suppose a ping-pong ball and a bowling ball
are rolling toward you. Both have the same
momentum, and you exert the same force to
stop each. How do the time intervals to stop
them compare?
1. It takes less time to stop the ping-pong ball.
2. Both take the same time.
3. It takes more time to stop the ping-pong
ball.
Suppose a ping-pong ball and a bowling ball
are rolling toward you. Both have the same
momentum, and you exert the same force to
stop each. How do the distances needed to
stop them compare?
1. It takes a shorter distance to stop the pingpong ball.
2. Both take the same distance.
3. It takes a longer distance to stop the pingpong ball.
A cart moving at speed v collides with an
identical stationary cart on an airtrack, and
the two stick together after the collision. What
is their velocity after colliding?
1. v
2. 0.5 v
3. zero
4. –0.5 v
5. –v
6. need more information
A person attempts to knock down a large
wooden bowling pin by throwing a ball at it.
The person has two balls of equal size and
mass, one made of rubber and the other of
putty. The rubber ball bounces back, while the
ball of putty sticks to the pin. Which ball is
most likely to topple the bowling pin?
1. the rubber ball
2. the ball of putty
3. makes no difference
4. need more information
Think fast! You’ve just driven around a curve
in a narrow, one-way street at 25 mph when
you notice a car identical to yours coming
straight toward you at 25 mph. You have only
two options: hitting the other car head on or
swerving into a massive concrete wall, also
head on. In the split second before the impact,
you decide to
1. hit the other car.
2. hit the wall.
3. hit either one—it makes no difference.
4. consult your lecture notes.
If all three collisions in the figure shown
here are totally inelastic, which bring(s) the
car on the left to a halt?
1. I
2. II
3. III
4. I, II
5. I, III
6. II, III
7. all three
If all three collisions in the figure shown
are totally inelastic, which cause(s) the
most damage?
1. I
2. II
3. III
4. I, II
5. I, III
6. II, III
7. all three
A golf ball is fired at a bowling ball initially
at rest and bounces back elastically. Compared
to the bowling ball, the golf ball after
the collision has
1. more momentum but less kinetic energy.
2. more momentum and more kinetic energy.
3. less momentum and less kinetic energy.
4. less momentum but more kinetic energy.
5. none of the above
A golf ball is fired at a bowling ball initially
at rest and sticks to it. Compared to the bowling
ball, the golf ball after the collision has
1. more momentum but less kinetic energy.
2. more momentum and more kinetic energy.
3. less momentum and less kinetic energy.
4. less momentum but more kinetic energy.
5. none of the above
Suppose you are on a cart, initially at rest on
a track with very little friction. You throw
balls at a partition that is rigidly mounted on
the cart. If the balls bounce straight back as
shown in the figure, is the cart put in motion?
1. Yes, it moves to the right.
2. Yes, it moves to the left.
3. No, it remains in place.
A compact car and a large truck collide head
on and stick together. Which undergoes the
larger momentum change?
1. car
2. truck
3. The momentum change is the same for both
vehicles.
4. Can’t tell without knowing the final velocity
of combined mass.
A compact car and a large truck collide
head on and stick together. Which vehicle
undergoes the larger acceleration during
the collision?
1. car
2. truck
3. Both experience the same acceleration.
4. Can’t tell without knowing the final
velocity of combined mass.
Is it possible for a stationary object that is
struck by a moving object to have a larger
final momentum than the initial momentum
of the incoming object?
1. Yes.
2. No because such an occurrence would
violate the law of conservation of
momentum.
Two carts of identical inertial mass are put backto-back on a track. Cart A has a spring loaded
piston; cart B is entirely passive. When the
piston is released, it pushes against cart B, and
1. A is put in motion but B remains at rest.
2. both carts are set into motion, with A gaining
more speed than B.
3. both carts gain equal speed but in opposite
directions.
4. both carts are set into motion, with B gaining
more speed than A.
5. B is put in motion but A remains at rest.
Two carts are put back-to-back on a track.
Cart A has a spring-loaded piston; cart B,
which has twice the inertial mass of cart A, is
entirely passive. When the piston is released,
it pushes against cart B, and the carts move
apart. How do the magnitudes of the final
momenta and kinetic energies compare?
1. pA > pB, kA > kB
2. pA > pB, kA = kB
3. pA > pB, kA < kB
4. pA = pB, kA > kB
5. pA = pB, kA = kB
Two carts are put back-to-back on a track.
Cart A has a spring-loaded piston; cart B,
which has twice the inertial mass of cart A,
is entirely passive. When the piston is released,
it pushes against cart B, and the carts
move apart. Ignoring signs, while the piston
is pushing,
1. A has a larger acceleration than B.
2. the two have the same acceleration.
3. B has a larger acceleration than A.
Two people on roller blades throw a ball back
and forth. Which statement(s) is/are true?
A. The interaction mediated by the ball is
repulsive.
B. If we film the action and play the movie
backward, the interaction appears attractive.
C. The total momentum of the two people
is conserved.
D. The total energy of the two people is
conserved.
What about the conservation laws? The ball carries both momentum and
energy back and forth between the two roller-bladers. Their momentum
and energy therefore cannot be conserved.
In the following figure, a 10-kg weight is
suspended from the ceiling by a spring. The
weight-spring system is at equilibrium with the
bottom of the weight about 1 m above the floor.
The spring is then stretched until the weight is
just above the eggs. When the spring is released,
the weight is pulled up by the contracting spring
and then falls back down under the influence of
gravity. On the way down, it
1. reverses its direction of travel well above the
eggs.
2. reverses its direction of travel precisely as it
reaches the eggs.
3. makes a mess as it crashes into the eggs.
In part (a) of the figure, an air track cart attached
to a spring rests on the track at the position xequilibrium
and the spring is relaxed. In (b), the cart is pulled to
the position xstart and released. It then oscillates about
xequilibrium.
Which graph correctly
represents the potential
energy of the spring as a
function of the
position of the cart?
Chapter 10:Rotation of a rigid object about a fixed axis
Part 1
Reading assignment: Chapter 10.6-10.9 (know concept of
moment of inertia, don’t worry about integral calculation)
Homework : (due Wednesday, Oct. 14, 2009):
Chapter 10:
Q23, 35, 48, 57, 80
• Rotational motion,
• Angular displacement, angular velocity, angular acceleration
• Rotational energy
• Moment of Inertia (Rotational inertia)
• Torque
• For every rotational quantity, there is a linear analog.
1. The center of mass of a rigid object of arbitrary shape
___ 1. is always inside the object.
___ 2. can lie outside the object.
___ 3. depends on the motion of the object.
___ 4. depends on the frame of reference of the object.
2. Compared with the kinetic energy of its center of mass (CM), the total
kinetic energy of a system is
___ 1. always less than the kinetic energy of the CM.
___ 2. always equal to the kinetic energy of the CM.
___ 3. greater than or equal to the kinetic energy of the CM.
___ 4. depends on the particular system
3. A rocket is propelled forward by ejecting gas at high speed. The
forward motion is a consequence of
___ 1. conservation of energy.
___ 2. conservation of momentum.
___ 3. both of the above.
___ 4. neither of the above.
Rotational motion
Look at one point P:
Arc length s:
s  r 
Thus, the angular
position is:
s

r
Planar, rigid object rotating about origin O.
 is measured in degrees or radians (more common)
r
s=r
One radian is the angle
subtended by an arc length
equal to the radius of the arc.

For full circle:  
Full circle has an angle of 2 radians,
Thus, one radian is 360°/2  57.3
s 2r

 2
r
r
Radian
degrees
2
360°

180°
/2
90°
1
57.3°
Define quantities for circular motion
(note analogies to linear motion!!)
Angular displacement:
D   f   i
 f i
D

Average angular speed:  
t f  ti
Dt
D d
Instantaneous angular speed:   lim

Dt 0 Dt
dt
Average angular acceleration:

Instantaneous angular acceleration:
 f  i
t f  ti

D
Dt
D d
  lim

Dt 0 Dt
dt
A ladybug sits at the outer edge of a merry-goround, and a gentleman bug sits halfway
between her and the axis of rotation. The
merry-go-round makes a complete revolution
once each second. The gentleman bug’s
angular speed is
1. half the ladybug’s.
2. the same as the ladybug’s.
3. twice the ladybug’s.
4. impossible to determine
Angular quantities
are vectors
Angular velocity, angular acceleration, angular
momentum, torque.
Right-hand rule for
determining the direction
of this vector.
Every particle (of a rigid object):
• rotates through the same angle,
• has the same angular velocity,
• has the same angular acceleration.
, ,  characterize rotational
motion of entire object
A ladybug sits at the outer edge of a merry-goround, that is turning and slowing down.
At the instant shown in the figure, the radial
component of the ladybug’s (Cartesian)
acceleration is
1. in the +x direction.
2. in the –x direction.
3. in the +y direction.
4. in the –y direction.
A ladybug sits at the outer edge of a merry-goround, that is turning and slowing down.
At the instant shown in the figure, the radial
component of the ladybug’s (Cartesian)
acceleration is
1. in the +x direction.
2. in the –x direction.
3. in the +y direction.
4. in the –y direction.
A ladybug sits at the outer edge of a merry-goround that is turning and is slowing down.
The vector expressing her angular velocity is
1. in the +x direction.
2. in the –x direction.
3. in the +y direction.
4. in the –y direction.
5. in the +z direction.
Linear motion with constant
linear acceleration, a.
Rotational motion with constant
rotational acceleration, .
v xf  v xi  a x t
 f   i  t
x f  xi  12 (vxi  vxf )t
 f   i  12 (i   f )t
1 2
x f  xi  v xi t  a x t
2
1 2
 f   i   i t  t
2
vxf  vxi  2ax ( x f  xi )
 f  i  2 ( f   i )
2
2
2
2
Black board example 11.1
A wheel starts from rest and
rotates with constant angular
acceleration and reaches an
angular speed of 12.0 rad/s in
3.00 s.
(a) What is the magnitude of the angular acceleration of the
wheel?
(b) Through what angle does the wheel rotate in these 3.00 s?
(c) Through which angle does the wheel rotate between t = 2.00 s
and 3.00 s?
Relation between linear
and angular quantities
Caution: Measure angular quantities in radians
Arc length s:
s  r 
Tangential speed of a
point P:
v  r 
Tangential acceleration of a
point P:
a  r 
Note, this is not the centripetal
acceleration ar !!
Black board example 11.2
The diameters of the main rotor
and the tail rotor of a
helicopter are 7.60 m and
1.02 m, respectively. The
respective rotational speeds
are 450 rev/min and 4138
rev/min.
a) Calculate the speeds of the
tips of both rotors.
b) Compare with the speed of
sound, 343 m/s.
c) The rotors are rotating at constant angular speed. What is the
centripetal acceleration and what is the angular acceleration?
Black board example 11.3
HW 27
(a) What is the angular speed 
about the polar axis of a
point on Earth’s surface at a
latitude of 40°N
(b) What is the linear speed v of
that point?
(c) What are  and v for a point
on the equator?
Radius of earth: 6370 km
1
2
1
2
Two wheels initially at rest roll the same distance
without slipping down identical inclined
planes starting from rest. Wheel B has
twice the radius but the same mass as wheel
A. All the mass is concentrated in their rims,
so that the rotational inertias are I = mR2.
Which has more translational kinetic energy
when it gets to the bottom?
1. Wheel A
2. Wheel B
3. The kinetic energies are the same.
4. need more information
Consider two people on opposite sides of a
rotating merry-go-round. One of them throws
a ball toward the other. In which frame of reference is the path of the ball straight when
viewed from above: (a) the frame of the
merry-go-round or (b) that of Earth?
1. (a) only
2. (a) and (b)—although the paths appear
to curve
3. (b) only
4. neither; because it’s thrown while in circular
motion, the ball travels along a
curved path.
You are using a wrench and trying to loosen
a rusty nut. Which of the arrangements
shown is most effective in loosening the nut?
List in order of descending efficiency the
following arrangements:
a) 1, 3, 4, 2
b) 2, 4, 1, 3
c) 4=2, 1, 3
d) 4=2, 1=3
e) 2, 1=4, 3
A force F is applied to a dumbbell for a time
interval Δt, first as in (a) and then as in (b). In
which case does the dumbbell acquire the
greater center-of-mass speed?
1. (a)
2. (b)
3. no difference
4. The answer depends on the rotational
inertia of the dumbbell.
Aforce F is applied to a dumbbell for a time
interval Δt, first as in (a) and then as in (b). In
which case does the dumbbell acquire the
greater energy?
1. (a)
2. (b)
3. no difference
4. The answer depends on the rotational
inertia of the dumbbell.
Imagine hitting a dumbbell with an object
coming in at speed v, first at the center, then
at one end. Is the center-of-mass speed of
the dumbbell the same in both cases?
1. yes
2. no
A box, with its center-of-mass off-center as indicated
by the dot, is placed on an inclined
plane. In which of the four orientations shown,
if any, does the box tip over?
Consider the situation shown at left below.
A puck of mass m, moving at speed v hits an
identical puck which is fastened to a pole
using a string of length r. After the collision,
the puck attached to the string revolves
around the pole. Suppose we now lengthen
the string by a factor 2, as shown on the
right, and repeat the experiment. Compared
to the angular speed in the first situation, the
new angular speed is
1. twice as high
2. the same
3. half as much
4. none of the above
A figure skater stands on one spot on the ice
(assumed frictionless) and spins around with
her arms extended. When she pulls in her
arms, she reduces her rotational inertia and
her angular speed increases so that her angular
momentum is conserved. Compared to her
initial rotational kinetic energy, her rotational
kinetic energy after she has pulled in her
arms must be
1. the same.
2. larger because she’s rotating faster.
3. smaller since her rotational inertia is smaller.
Two wheels with fixed hubs, each having a
mass of 1 kg, start from rest, and forces are
applied as shown. Assume the hubs and
spokes are massless, so that the rotational inertia
is I = mR2. In order to impart identical
angular accelerations, how large must F2 be?
1. 0.25 N
2. 0.5 N
3. 1 N
4. 2 N
5. 4 N
Consider the uniformly rotating object
shown below. If the object’s angular velocity
is a vector (in other words, it points in a
certain direction in space) is there a particular
direction we should associate with the
angular velocity?
1. yes, ±x
2. yes, ±y
3. yes, ±z
4. yes, some other direction
5. no, the choice is really arbitrary
A person spins a tennis ball on a string in a
horizontal circle (so that the axis of rotation
is vertical). At the point indicated below, the
ball is given a sharp blow in the forward
direction. This causes a change in angular
momentum ΔL in the
1. x direction
2. y direction
3. z direction
A person spins a tennis ball on a string in a
horizontal circle (so that the axis of rotation is
vertical). At the point indicated below, the ball
is given a sharp blow vertically downward. In
which direction does the axis of rotation tilt
after the blow?
1. +x direction
2. –x direction
3. +y direction
4. –y direction
5. It stays the same (but the magnitude of
the angular momentum changes).
6. The ball starts wobbling in all directions.
A suitcase containing a spinning flywheel is
rotated about the vertical axis as shown in
(a). As it rotates, the bottom of the suitcase
moves out and up, as in (b). From this, we
can conclude that the flywheel, as seen from
the side of the suitcase as in (a), rotates
1. clockwise.
2. counterclockwise.
Chapter 10:Rotation of a rigid object about a fixed axis
Part 2
Reading assignment: Chapter 11.1-11.3
Homework : (due Wednesday, Oct. 14, 2009):
Problems:
Chapter 11 9AE's, 3AF's, Q2, 9, 16, 26, 28
• Rotational motion,
• Angular displacement, angular velocity, angular acceleration
• Rotational energy
• Moment of Inertia (Rotational inertia)
• Torque
• For every rotational quantity, there is a linear analog.
1. An object is rotated about a vertical axis by 90° and then about a horizontal axis by
180°. If we start over and perform the rotations in the reverse order, the orientation of
the object
___ 1. will be the same as before.
___ 2. will be different than before.
___ 3. depends on the shape of the object.
2. A disk is rotating at a constant rate about a vertical axis through its center. Point Q is
twice as far from the center of the disk as point P is. The angular velocity of Q at a
given time is
___ 1. twice as big as P’s.
___ 2. the same as P’s.
___ 3. half as big as P’s.
___ 4. none of the above.
3. When a disk rotates counterclockwise at a constant rate about a vertical axis through
its center, the tangential acceleration of a point on the rim is
___ 1. positive.
___ 2. zero.
___ 3. negative.
___ 4. impossible to determine without more information.
Rotational
energy
A rotating object (collection
of i points with mass mi) has
a rotational kinetic energy of
1
2
K R  I 
2
Where:
I   mi  ri
i
2
Rotational inertia
Demo:
Both sticks have the same weight.
Why is it so much more difficult to
rotate the blue stick?
Black board example 11.4
2
What is the
rotational inertia?
3
1
4
Four small spheres are mounted on the corners of a frame as shown.
a) What is the rotational energy of the system if it is rotated about
the z-axis (out of page) with an angular velocity of 5 rad/s
b) What is the rotational energy if the system is rotated about the yaxis?
(M = 5 kg; m = 2 kg; a = 1.5 m; b = 1 m).
Rotational inertia of an object depends on:
- the axis about which the object is rotated.
- the mass of the object.
- the distance between the mass(es) and the axis
of rotation.
I   mi  ri
i
2
Calculation of Rotational inertia for
continuous extended objects
I
lim  ri
Dmi 0 i
2
 Dmi   r dm   r dV
2
2
Refer to Table11-2
Note that the moments of inertia are different for different axes
of rotation (even for the same object)
1
I  ML2
3
I
1
ML2
12
1
I  MR 2
2
Rotational inertia for some objects
Page 278
Parallel axis
theorem
 Rotational inertia for a rotation about an axis that is
parallel to an axis through the center of mass
I CM
I  I CM  Mh 2
h
Blackboard example 11.4
What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is
rotating about an axis 0.5 away from the center with  = 2 rad/sec?
Conservation of energy (including rotational energy):
Again:
If there are no non-conservative forces: Energy is conserved.
Rotational kinetic energy must be included in energy
considerations!
Ei  E f
U i  Klinear,initial  K rotational,initial  U f  Klinear, final  K rotational, final
Black board example 11.5
Connected cylinders.
Two masses m1 (5 kg) and m2 (10
kg) are hanging from a pulley of
mass M (3 kg) and radius R (0.1
m), as shown. There is no slip
between the rope and the pulleys.
(a) What will happen when the
masses are released?
(b) Find the velocity of the masses after they have fallen a
distance of 0.5 m.
(c) What is the angular velocity of the pulley at that moment?
Torque
F  sin f
r
F
f
F  cos f
A force F is acting at an angle f on a lever that is rotating around
a pivot point. r is the distance between F and the pivot point.
This force-lever pair results in a torque t on the lever
t  r  F  sin f
Black board example 11.6
Two mechanics are trying to
open a rusty screw on a ship
with a big ol’ wrench. One
pulls at the end of the wrench
(r = 1 m) with a force F = 500
N at an angle F1 = 80 °; the
other pulls at the middle of
wrench with the same force
and at an angle F2 = 90 °.
What is the net torque the two mechanics are applying to the screw?
Torque t and
angular acceleration .
Newton’s 2. law for rotation.
Particle of mass m rotating in a
circle with radius r.
Radial force Fr to keep particle on
circular path.
Tangential force Ft accelerates
particle along tangent.
Ft  mat
Torque acting on particle is proportional to
angular acceleration :
t  I
 
dW  F  ds
 
W  F s
Definition of work:
Work in linear motion:
 
dW  F  ds
 
W  F  s  F  s  cos 
Component of force F along
displacement s. Angle 
between F and s.
Work in rotational motion:
 
dW  F  ds
Torque t and angular
dW  t  d
W  t 
displacement .
Work and Energy in rotational motion
Remember work-kinetic energy theorem for linear motion:
1
1
2
2
W

mv

mv

f
i
2
2
External work done on an object changes its kinetic energy
There is an equivalent work-rotational kinetic energy theorem:
1
1
2
2
 W  I f  I i
2
2
External, rotational work done on an object changes its rotational kinetic energy
Linear motion with constant
linear acceleration, a.
Rotational motion with constant
rotational acceleration, .
v xf  v xi  a x t
 f   i  t
x f  xi  12 (vxi  vxf )t
 f   i  12 (i   f )t
1 2
x f  xi  v xi t  a x t
2
1 2
 f   i   i t  t
2
vxf  vxi  2ax ( x f  xi )
 f  i  2 ( f   i )
2
2
2
2
Announcements
1. Midterm 2 on Wednesday, Oct. 21.
2. Material: Chapters 7-11
3. Review on Tuesday (outside of class time)
4. I’ll post practice tests on Web
5. You are allowed a 3x5 inch cheat card with 10 equations
6. Go through practice exams & homework & class
examples; understand concepts & demos
7. Time limit for test: 50 minutes
1. The rotational inertia of a rigid body
___ 1. is a measure of its resistance to changes in rotational motion.
___ 2. depends on the location of the axis of rotation.
___ 3. is large if most of the body’s mass is far from the axis of rotation.
___ 4. is all of the above.
___ 5. is none of the above.
2. The angular momentum of a particle
___ 1. is independent of the specific origin of coordinates.
___ 2. is zero when its position and momentum vectors are parallel.
___ 3. is zero when its position and momentum vectors are perpendicular.
___ 4. is not covered in the reading assignment.
3. A wheel rolls without slipping along a horizontal surface. The center of
the wheel has a translational speed v. The lowermost point on the wheel
has a net forward velocity
___ 1. 2v.
___ 2. v.
___ 3. zero.
___ 4. need more information
Linear motion with constant
linear acceleration, a.
Rotational motion with constant
rotational acceleration, .
v xf  v xi  a x t
 f   i  t
x f  xi  12 (vxi  vxf )t
 f   i  12 (i   f )t
1 2
x f  xi  v xi t  a x t
2
1 2
 f   i   i t  t
2
vxf  vxi  2ax ( x f  xi )
 f  i  2 ( f   i )
2
2
2
2
Summary: Angular and linear quantities
Linear motion
Rotational motion
Kinetic Energy:
1
K  m  v2
2
Kinetic Energy:
1
K R  I  2
2
Force:
F  ma
Torque:
t  I
Momentum:
p  mv
Work:
 
W  F s
Angular Momentum:
Work:
L  I
W  t 
Rolling motion
Pure rolling:
There is no slipping
Linear speed of center of mass:
vCM
ds R  d


 R 
dt
dt
Rolling motion
The angular velocity of
any point on the wheel is
the same.
The linear speed of any point on the object changes as shown in the
diagram!!
For one instant (bottom), point P has no linear speed.
For one instant (top), point P’ has a linear speed of 2·vCM
Rolling motion of a particle on a wheel
(Superposition of rolling and linear motion)
Rolling
=
Rotation
+
Linear
Rolling motion
Superposition principle:
Rolling motion
=
Kinetic energy
of rolling motion:
Pure translation +
Pure rotation
1
1
2
2
K  Mv  I CM  
2
2
Chapter 11: Angular Momentum part 1
Reading assignment: Chapter 11.4-11.6
Homework : (due Monday, Oct. 17, 2005):
Problems:
30, 41, 42, 44, 48, 53
• Torque
• Angular momentum
• Angular momentum is conserved
Torque and the vector product
Thus far:
Torque
t  r  F  sin F
Torque is the vector product
between the force vector F
and vector r
 
t  r F

Torque and the vector product
Definition of vector product:
f
  
C  A B
- The vector product of vectors A and B is the vector C.
- C is perpendicular to A and B
- The magnitude of C = A·B·sinf
Torque and the vector product
  
C  A B
f
Use the right hand rule to figure out the direction of C.
- Thumb is C (i.e. torque t, angular velocity , angular momentum L)
- Index finger is A (e.g. radius r)
- Middle finger is B (e.g. force F)
Torque and the vector product
  
C  A B
f
Rules for the vector product.
 
 
1. A  B   B  A
 
 
Thus, A  A  0
2. If A is parallel to B then A  B  0.
 
3. If A is perpendicular to B then A  B  A  B
  
   
4. A  ( B  C )  A  B  A  C
5. Magnitude of C = A·B·sin is equal to area of parallelogram
made by A and B
Torque and the vector product
  
C  A B
f
Rules for the vector product (cont).

 


6. A  B  ( Ay Bz  Az B y )i  ( Ax Bz  Az Bx ) j  ( Ax B y  Ay Bx )k
Black board example 12.2
A force F = (2.00i + 3.00j) is
applied to an object that is
pivoted about a fixed axis
aligned along the z-axis.
The force is applied at the point
r = (4.00i + 5.00j).
(a) What is the torque exerted on the object?
(b) What is the magnitude and direction of the torque vector t.
(c) What is the angle between the directions of F and r?
Angular momentum of a particle
Definition:
  
Lrp
 
 m( r  v )
L… angular momentum
r… distance from the origin
p… momentum of particle
v…velocity of particle
L is perpendicular to r and p
L has magnitude L = r·p·sinF
Angular momentum of a rotating
rigid object
We’ll consider an object that is
rotating about the z-axis.
The angular momentum of the
object is given by:
Lz  I  
Note that in this case L and  are along the z axis.
Also note the analog formula for linear momentum p = m·v
Black board example 12.3
A light rigid rod, 1 m in length,
joins two particles – with
masses 3 kg and 4 kg at its end.
The system rotates in the x-y
plane about a pivot through the
center of the rod.
Determine the angular
momentum of the system about
the origin when the speed of
each particle is 5.00 m/s.
Conservation of angular momentum
The total angular momentum of a system is constant in both
magnitude and direction if the resultant external torque
acting on the system is zero.

L  constant
If the system undergoes an internal “rearrangement”, then
 
Li  L f  constant
If the object is rotating about a fixed axis (say z-axis), then:
I i i  I f  f  constant
Conservation laws
Ki  U i  K f  U f 

 
pi  p f
 For an isolated system
 

Li  L f

Demo
A students stands still on a rotatable platform
and holds a spinning wheel. The bicycle wheel is
spinning in the clockwise direction when viewed
from above.
He flips the wheel over.
What happens?
Black board example 12.4
Student on a turn table.
A student stands on a platform that is rotating with an angular
speed of 7.5 rad/s, his arms outstretched and he holds a brick
in each hand. The rotational inertia of the whole system is 6.0
kg·m2. The student then pulls the bricks inward thus reducing
the rotational inertia to 2.0 kg·m2.
(a) What is the new angular speed of the platform?
(b) What is the ratio of the new kinetic energy of the system to
the original kinetic energy?
(c) What provided the added kinetic energy?
Summary: Angular and linear quantities
Linear motion
1
2
K

m

v
Kinetic Energy:
2
Force:
F  ma
Momentum:
p  mv
Work:
 
W  F s
Rotational motion
Kinetic Energy:
Torque:
Angular Momentum:
Work:
1
K R  I  2
2
t  I
L  I
W  t 