Goals:

1
Lecture 11

• Goals

• Chapter 9 Momentum Impulse
• Understand what momentum is and how it relates
  to forces
• Employ momentum conservation principles
• In problems with 1D and 2D Collisions
• In problems having an impulse (Force vs. time)
• Chapter 8 Use models with free fall

• Assignment
• Read through Chapter 10
• MP HW5, due Wednesday 3/3

2
Problem 7.34 Hint

• Suggested Steps
• Two independent free body diagrams are necessary
• Draw in the forces on the top and bottom blocks
• Top Block
• Forces  1. normal to bottom block 2. weight 3.
  rope tension and 4. friction with bottom
  block (model with sliding)
• Bottom Block
• Forces 
• 1. normal to bottom surface
• 2. normal to top block interface
• 3. rope tension (to the left)
• 4. weight (2 kg)
• 5. friction with top block
• 6. friction with surface
• 7. 20 N
• Use Newton's 3rd Law to deal with the force pairs
  
• (horizontal vertical) between the top and
  bottom block.

3
Locomotion how fast can a biped walk?

4
How fast can a biped walk?

• What about weight?
• A heavier person of equal height and proportions
  can walk faster than a lighter person
• A lighter person of equal height and proportions
  can walk faster than a heavier person
• To first order, size doesnt matter

5
How fast can a biped walk?

• What about height?
• A taller person of equal weight and proportions
  can walk faster than a shorter person
• A shorter person of equal weight and proportions
  can walk faster than a taller person
• To first order, height doesnt matter

6
How fast can a biped walk?
What can we say about the walkers acceleration
if there is UCM (a smooth walker) ?
Acceleration is radial !
So where does it, ar, come from? (i.e., what
external forces are on the walker?)
1. Weight of walker, downwards 2. Friction with
the ground, sideways
7
Orbiting satellites vT (gr)½
8
Geostationary orbit
9
Geostationary orbit

• The radius of the Earth is 6000 km but at 36000
  km you are 42000 km from the center of the
  earth.
• Fgravity is proportional to r-2 and so little g
  is now 10 m/s2 / 50
• vT (0.20 42000000)½ m/s 3000 m/s
• At 3000 m/s, period T 2p r / vT 2p 42000000
  / 3000 sec
• 90000 sec 90000 s/ 3600 s/hr 24 hrs
  
• Orbit affected by the moon and also the Earths
  mass is inhomogeneous (not perfectly
  geostationary)
• Great for communication satellites
• (1st pointed out by Arthur C. Clarke)

10
Impulse Linear Momentum

• Transition from forces to conservation laws
• Newtons Laws ? Conservation Laws
• Conservation Laws ? Newtons Laws
• They are different faces of the same physics
• NOTE We have studied impulse and momentum
  but we have not explicitly named them as such
• Conservation of momentum is far more general than
• conservation of mechanical energy

11
Collisions are a fact of life
12
Forces vs time (and space, Ch. 10)

• Underlying any new concept in Chapter 9 is
• A net force changes velocity (either magnitude
  or direction)
• For any action there is an equal and opposite
  reaction
• If we emphasize Newtons 3rd Law and look at the
  changes with time then this leads to the
  Conservation of Momentum Principle

13
Example 1

• A 2 kg block, initially at rest on frictionless
  horizontal surface, is acted on by a 10 N
  horizontal force for 2 seconds (in 1D).
• What is the final velocity?
• F is to the positive F ma thus a F/m 5
  m/s2
• v v0 a Dt 0 m/s 2 x 5 m/s 10 m/s (
  direction)
• Notice v - v0 a Dt ? m (v - v0) ma Dt ? m
  Dv F Dt
• If the mass had been 4 kg now what final
  velocity?

14
Twice the mass
Before

• Same force
• Same time
• Half the acceleration (a F / m)
• Half the velocity ! ( 5 m/s )

0
2
Time (sec)
15
Example 1

• Notice that the final velocity in this case is
  inversely proportional to the mass (i.e., if
  thrice the mass.one-third the velocity).
• Here, mass times the velocity always gives the
  same value. (Always 20 kg m/s.)

Area under curve is still the same ! Force x
change in time mass x change in velocity
16
Example 1

• There many situations in which the sum of the
  products mass times velocity is constant over
  time
• To each product we assign the name, momentum
  and associate it with a conservation law.
  
• (Units kg m/s or N s)
• A force applied for a certain period of time can
  be graphed and the area under the curve is the
  impulse

Area under curve impulse With m Dv Favg Dt
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Force curves are usually a bit different in the
real world
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Example 1 with Action-Reaction

• Now the 10 N force from before is applied by
  person A on person B while standing on a
  frictionless surface
• For the force of A on B there is an equal and
  opposite force of B on A

MA x DVA Area of top curve MB x DVB Area
of bottom curve Area (top) Area (bottom) 0
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Example 1 with Action-Reaction

• MA DVA MB DVB 0
• MA VA(final) - VA(initial) MB VB(final) -
  VB(initial) 0
• Rearranging terms

MAVA(final) MB VB(final) MAVA(initial) MB
VB(initial) which is constant regardless of M or
DV (Remember frictionless surface)
20
Example 1 with Action-Reaction
MAVA(final) MB VB(final) MAVA(initial) MB
VB(initial) which is constant regardless of M or
DV
Define MV to be the momentum and this is
conserved in a system if and only if the system
is not acted on by a net external force (choosing
the system is key) Conservation of momentum is
a special case of applying Newtons Laws
21
Applications of Momentum Conservation
Radioactive decay
Explosions
Collisions
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Impulse Linear Momentum

• Definition For a single particle, the momentum
  p is defined as

p mv
(p is a vector since v is a vector)
So px mvx and so on (y and z directions)

• Newtons 2nd Law

F ma

• This is the most general statement of Newtons
  2nd Law

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Momentum Conservation

• Momentum conservation (recasts Newtons 2nd Law
  when F 0) is an important principle
• It is a vector expression (Px, Py and Pz) .
• And applies to any situation in which there is
  NO net external force applied (in terms of the x,
  y z axes).

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Momentum Conservation

• Many problems can be addressed through momentum
  conservation even if other physical quantities
  (e.g. mechanical energy) are not conserved
• Momentum is a vector quantity and we can
  independently assess its conservation in the x, y
  and z directions
• (e.g., net forces in the z direction do not
  affect the momentum of the x y directions)

25
Exercise 1 Momentum is a Vector (!) quantity

• A block slides down a frictionless ramp and then
  falls and lands in a cart which then rolls
  horizontally without friction
• In regards to the block landing in the cart is
  momentum conserved?

1. Yes
2. No
3. Yes No
4. Too little information given

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Exercise 1 Momentum is a Vector (!) quantity

• x-direction No net force so Px is conserved.
• y-direction Net force, interaction with the
  ground so
• depending on the system (i.e., do you include the
  Earth?)
• Py is not conserved (system is block and cart
  only)

2 kg
5.0 m

• Let a 2 kg block start at rest on a 30 incline
  and slide vertically a distance 5.0 m and fall a
  distance 7.5 m into the 10 kg cart
• What is the final velocity of the cart?

30
10 kg
7.5 m
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Inelastic collision in 1-D Example 2

• A block of mass M is initially at rest on a
  frictionless horizontal surface. A bullet of
  mass m is fired at the block with a muzzle
  velocity (speed) v. The bullet lodges in the
  block, and the block ends up with a speed V. In
  terms of m, M, and V
• What is the momentum of the bullet with speed v ?

x
v
V
before
after
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Inelastic collision in 1-D Example 2

• What is the momentum of the bullet with speed v
  ?
• Key question Is x-momentum conserved ?

Before
After
v
V
x
before
after
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Example 2Inelastic Collision in 1-D with numbers
Do not try this at home!
ice
(no friction)
Before A 4000 kg bus, twice the mass of the
car, moving at 30 m/s impacts the car at rest.
What is the final speed after impact if they
move together?
30
Exercise 2Momentum Conservation

• Two balls of equal mass are thrown horizontally
  with the same initial velocity. They hit
  identical stationary boxes resting on a
  frictionless horizontal surface.
• The ball hitting box 1 bounces elastically back,
  while the ball hitting box 2 sticks.
• Which box ends up moving fastest ?

1. Box 1
2. Box 2
3. same