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2/13/15 Conservation of Momentum
The principle of conservation of momentum
states: When there is no net external force
on a system of objects, the total momentum
of the system does not change.
We say that the momentum of such a
system is conserved since it does not
change.
Momentum
Object A
after collision
Impact
After Collision
Demo: Elastic vs. Inelastic
Collisions
•  The change of momentum of the two
objects in a collision is equal and opposite
-- the momentum gained by one object is
the amount lost by the other.
Momentum
Object B
Before collision
Before Collision
•  Each object has equal and opposite change in
momentum.
•  The momentum of the system of 2 objects is
conserved.
Collisions
Momentum
Object A
before collision
Collisions
•  When two objects collide, impulse is equal and
opposite for the two objects if there is no net
external force.
Momentum
Object B
after collision
Collisions
A sharp dart and a blunt dart will each be fired
at the wooden block.
The sharp dart will stick in the block, and the
blunt dart will bounce off.
Will they exert the same impulse on the wooden
block? If not, which one will exert more
impulse?
Recoil: Also explained by
conservation of momentum
1. Elastic collision: Objects rebound without
deforming or generating heat
–  Billiard balls
2. Inelastic: objects hit, but deform and/or generate
heat
–  Tennis ball being hit by a racket
Mxv
mxV
3. Perfectly inelastic: Objects hit and stick together
•  Momentum is conserved, whether collision is
elastic or inelastic, as long as there’s no net
external force!
1 2/13/15 Main Points:
Example: Big fish eats smaller fish!
If the big fish has a mass 5x bigger than the
small fish, what is the speed of the big fish
after lunch?
Chapter 7: Work and Energy
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Definition of Momentum
Definition of Impulse
Need an impulse to change momentum
Conservation of Momentum in collisions
Different types of collisions
Work: The transfer of energy
(something must be moved)
•  Definition of work done on an object: Work =
force x distance traveled
•  W = F d
•  SI Unit of work: Joule (J), 1 = 1 N!m
•  d is distance along force direction
–  Force acting in direction of motion: Positive work.
–  Force acting in opposite direction: Negative work.
–  Force perpendicular to motion: Zero work
Examples: Work
Work against & by Gravity
•  Imagine dragging a heavy box along the
ground. What forces act on the box?
Which forces do work? Do they do
positive work, or negative work?
2 2/13/15 Examples: Loading a Ship
•  3,000 kg truck is loaded onto a
ship by crane that exerts
upward force of 31,000 N on
truck. This force is applied over
a distance of 2.0 m.
(a) Find work done on truck by
crane
(b) Find work done on truck by
gravity.
(c) Find net work done on the truck.
Typical Power Outputs
Human basal
metabolism is
around 80 W
Mechanical Energy: Energy due to
position or movement
Power
•  Power is a measure of the rate at which
work is done. If work W done during time t:
P = Work
time
•  SI unit: 1 J/s = 1 watt = 1 W
•  Horsepower: 1 hp = 746 W
Example: Power
•  How much power is expended when a
weightlifter lifts a 500 N barbell 2.2 m in 2
seconds?
Gravitational Potential Energy
•  Potential Energy: stored energy
(gravitational, spring, etc.)
–  Gravitational PE = mgh
•  Kinetic Energy: energy associated with
motion
–  KE = ½ mv2
•  Both have units of Joules!
3 2/13/15 Example: Kinetic Energy
•  A person pushes a 10 kg crate so that it
accelerates up to a speed of 10 m/s.
•  What is the final kinetic energy of the
crate?
•  Where did this kinetic energy come from?
Work-Kinetic Energy Theorem
•  By causing an object to speed up or slow down, you
change its kinetic energy.
•  The change in kinetic energy EQUALS the amount of
work that you did!
W = ΔKE
Careful: Work is not the same thing as energy,
it’s a way of transferring energy from place to
place or from one form into another form
Example: Rollercoasters
Ranking: Which has the greatest
kinetic energy?
•  Where is the cart going the fastest?
•  Could the cart ever get higher than
point A on its own?
Ranking: A ball is released from rest,
and rolls along a nearly frictionless
track. Rank the four points in terms of
their potential energy, from greatest to
least.
Ranking: A ball is released from rest,
and rolls along a nearly frictionless
track. Rank the four points in terms of
their kinetic energy, from greatest to
least.
4 2/13/15 Conservation of Energy
•  Energy cannot be created or destroyed; it
may be transformed from one form into
another, but the total amount of energy
never changes
Example: Free Fall revisited
•  If you drop a ball from height of 100 m,
how fast is it going right before it hits the
ground?
•  How much time did it spend falling?
Rotational Motion
Chapter 8: Rotational Motion
In physics we distinguish two types of
motion for objects:
•  Translational Motion (change of location):
Whole object moves through space.
•  Rotational Motion - object turns around an
axis (axle); axis does not move. (Wheels)
Angular Position
•  We will measure
angular position in
revolutions:
•  Counterclockwise
(CCW): positive
rotation
•  Clockwise (CW):
negative rotation
Linear Distance d vs. Angular
distance Δθ
For a point at
radius R on
the wheel,
d = 2πRΔθ
for Δθ in
revolutions
R
5 2/13/15 Ranking: Rolling Cups
•  Which of the cups will roll in the straightest path?
•  Which of the cups will roll in the most curved path?
Angular Velocity: ω
•  Avg. Angular Velocity = # Revolutions/
(Time Taken)
ω = Δθ / t
•  Unit: Revolutions/s or Revolutions/min
(RPM)
•  Sign convention : ω is positive for
counterclockwise rotation, negative for
clockwise rotation
Tangential Velocity
•  Every spot on a rotating object has both
angular velocity and tangential velocity.
•  vt = Δd/Δt = 2πRΔθ/Δt = 2πRω
Speed in Circular Motion
•  Rotational Speed ω: Rev.s per second
•  Tangential speed vt: distance per second
•  Two objects can have the same rotational
speed, but different tangential speeds!
RΔθ
R
Example: Gears
•  Two wheels are
connected by a chain
that doesn’t slip.
•  Which wheel has the
higher rotational
speed?
•  Which wheel has the
higher tangential speed
for a point on its rim?
Angular Acceleration
•  Change in angular
velocity -> angular
acceleration!
•  However, even if
angular velocity is
constant, each point
also has centripetal
acceleration (due to
change in direction of
vt)
6 2/13/15 Simple vs. Complex Objects
Model motion with just
Position
Model motion with
position and Rotation
Rotational Inertia
•  Rotational inertia
depends on
–  Total mass of the object
–  Distribution of the mass
relative to axis
•  Farther the mass is from
the axis of rotation, the
larger the rotational
inertia.
•  Rotational inertia ~
(mass) x (axis_distance)2
Rotational Inertia
Depends upon the axis
around which it rotates
•  Easier to rotate pencil
around an axis passing
through it.
•  Harder to rotate it around
vertical axis passing
through center.
•  Hardest to rotate it around
vertical axis passing
through the end.
Example: Hoop vs. Disk
•  Imagine rolling a hoop
and a disk of equal
mass down a ramp.
Which one would
win?
•  Which one is “easier”
to rotate (i.e., has less
rotational inertia)?
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