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Understand what momentum is and how it relates to forces ... proportional to the mass (i.e., if thrice the mass....one-third the velocity) ... – PowerPoint PPT presentation

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Title: 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
17
Force curves are usually a bit different in the
real world
18
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
19
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
22
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

23
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).

24
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

26
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
27
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
28
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
29
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
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