Physics%20207:%20Lecture%202%20Notes

1
Lecture 15

• Goals
• Employ conservation of momentum in 1 D 2D
• Introduce Momentum and Impulse
• Compare Force vs time to Force vs distance
• Introduce Center-of-Mass
• Note 2nd Exam, Monday, March 19th, 715 to 845
  PM

2
Comments on Momentum Conservation

• More general than conservation of mechanical
  energy
• Momentum Conservation occurs in systems with no
  net external forces (as a vector quantity)

3
Explosions A collision in reverse

• A two piece assembly is hanging vertically at
  rest at the end of a 20 m long massless string.
  The mass of the two pieces are 60 and 20 kg
  respectively. Suddenly you observe that the 20
  kg is ejected horizontally at 30 m/s. The time
  of the explosion is short compared to the swing
  of the string.
• Does the tension in the string increase or
  decrease after the explosion?
• If the time of the explosion is short then
  momentum is conserved in the x-direction because
  there is no net x force. This is not true of the
  y-direction but this is what we are interested in.

After
Before
4
Explosions A collision in reverse

• A two piece assembly is hanging vertically at
  rest at the end of a 20 m long massless string.
  The mass of the two pieces are 60 and 20 kg
  respectively. Suddenly you observe that the 20
  kg mass is ejected horizontally at 30 m/s.
• Decipher the physics
• 1. The green ball recoils in the x direction
  (3rd Law) and, because there is no net external
  force in the x-direction the x-momentum is
  conserved.
• 2. The motion of the green ball is constrained
  to a circular paththere must be centripetal
  (i.e., radial acceleration)

After
Before
5
Explosions A collision in reverse

• A two piece assembly is hanging vertically at
  rest at the end of a 20 m long massless string.
  The mass of the two pieces are 60 20 kg
  respectively. Suddenly you observe that the 20
  kg mass is suddenly ejected horizontally at 30
  m/s.
• Cons. of x-momentum
• px before px after 0 - M V m v
• V m v / M 2030/ 60 10 m/s
• Tbefore Weight (6020) x 10 N 800 N
• SFy m acy M V2/r T Mg
• T Mg MV2 /r 600 N 60x(10)2/20 N 900 N

After
6
Exercise 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

7
Exercise 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
30
10 kg

• 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?

7.5 m
8
Exercise Momentum is a Vector (!) quantity
1) ai g sin 30 5 m/s2 2) d 5 m
/ sin 30 ½ ai Dt2 10 m 2.5 m/s2
Dt2 2s Dt v ai Dt 10 m/s vx v
cos 30 8.7 m/s

• x-direction No net force so Px is conserved
• y-direction vy of the cart block will be zero
  and we can ignore vy of the block when it lands
  in the cart.

N
5.0 m
mg
30
30

• Initial Final
• Px MVx mvx (Mm) Vx
• M 0 mvx (Mm) Vx
• Vx m vx / (M m)
• 2 (8.7)/ 12 m/s
• Vx 1.4 m/s

7.5 m
y
x
9
Impulse (A variable external force applied for a
given time)

• Collisions often involve a varying force
• F(t) 0 ? maximum ? 0
• We can plot force vs time for a typical
  collision. The impulse, I, of the force is a
  vector defined as the integral of the force
  during the time of the collision.
• The impulse measures momentum transfer

10
Force and Impulse (A variable force applied for
a given time)

• J a vector that reflects momentum transfer

F
Impulse I area under this curve ! (Transfer of
momentum !)
Impulse has units of Newton-seconds
11
Force and Impulse

• Two different collisions can have the same
  impulse since I depends only on the momentum
  transfer, NOT the nature of the collision.

same area
F
t
?t
?t
?t big, F small
?t small, F big
12
Average Force and Impulse
Fav
F
Fav
t
?t
?t
?t big, Fav small
?t small, Fav big
13
Exercise Force Impulse

• Two boxes, one heavier than the other, are
  initially at rest on a horizontal frictionless
  surface. The same constant force F acts on each
  one for exactly 1 second.
• Which box has the most momentum after the force
  acts ?

1. heavier
2. lighter
3. same
4. cant tell

14
Discussion Exercise

• The only force acting on a 2.0 kg object moving
  along the x-axis. Notice that the plot is force
  vs time.
• If the velocity vx is 2.0 m/s at 0 sec, what is
  vx at 4.0 s ?
• Dp m Dv Impulse
• m Dv I0,1 I1,2 I2,4
• m Dv (-8)1 N s
• ½ (-8)1 N s ½ 16(2) N s
• m Dv 4 N s
• Dv 2 m/s
• vx 2 2 m/s 4 m/s

15
A perfectly inelastic collision in 2-D

• Consider a collision in 2-D (cars crashing at a
  slippery intersection...no friction).

V
v1
q
m1 m2
m1
m2
v2
before
after

• If no external force momentum is conserved.
• Momentum is a vector so px, py and pz

16
A perfectly inelastic collision in 2-D

• If no external force momentum is conserved.
• Momentum is a vector so px, py and pz are
  conseved

V
v1
m1 m2
q
m1
m2
v2
before
after

• x-dir px m1 v1 (m1 m2 ) V cos q
• y-dir py m2 v2 (m1 m2 ) V sin q

17
2D Elastic Collisions

• Perfectly elastic means that the objects do not
  stick and, by stipulation, mechanical energy is
  conservsed.
• There are many more possible outcomes but, if no
  external force, then momentum will always be
  conserved

18
Billiards

• Consider the case where one ball is initially at
  rest.

after
before
pa q
pb
vcm
Pa f
F
The final direction of the red ball will depend
on where the balls hit.
19
Billiards Without external forces, conservation
of both momentum mech. energy

• Conservation of Momentum
• x-dir Px m vbefore m vafter cos q m Vafter
  cos f
• y-dir Py 0 m vafter sin q m
  Vafter sin f

If the masses of the two balls are equal then
there will always be a 90 angle between the
paths of the outgoing balls
20
Center of Mass

• Most objects are not point-like but have a mass
  density and are often deformable.
• So how does one account for this complexity in a
  straightforward way?
• Example
• In football coaches often tell players attempting
  to tackle the ball carrier to look at their
  navel.
• So why is this so?

21
System of Particles Center of Mass (CM)

• If an object is not held then it will rotate
  about the center of mass.
• Center of mass Where the system is balanced !
• Building a mobile is an exercise in finding
• centers of mass.

mobile
22
System of Particles Center of Mass

• How do we describe the position of a system
  made up of many parts ?
• Define the Center of Mass (average position)
• For a collection of N individual point-like
  particles whose masses and positions we know

(In this case, N 2)
23
Momentum of the center-of-mass is just the total
momentum

• Notice

• Impulse and momentum conservation applies to the
  center-of-mass

24
Sample calculation

• Consider the following mass distribution

XCM (m x 0 2m x 12 m x 24 )/4m meters YCM
(m x 0 2m x 12 m x 0 )/4m meters XCM 12
meters YCM 6 meters
25
A classic example

• There is a disc of uniform mass and radius r.
  However there is a hole of radius a a distance b
  (along the x-axis) away from the center.
• Where is the center of mass for this object?

26
System of Particles Center of Mass

• For a continuous solid, convert sums to an
  integral.

dm
where dm is an infinitesimal mass element (see
text for an example).
r
y
x
27
Recap

• Thursday, Review for exam
• For Tuesday, Read Chapter 10.1-10.5