Download Chapter 7 - TESD home

Physics 3310
Giancoli
Chapter 7
Note Packet
Linear Momentum
_________________________
STUDENT NAME
_________________________
TEACHER’S NAME
______________
PERIOD
2
Physics 3310-XX
3
Physics 3310-XX
Chapter 7
Linear Momentum & Center of Mass
Definition: momentum = mass X velocity


p  mv
where

p  momentum
When we take the timed rate of change of

p


p
v

m
 ma
t
t
By Newton’s Second Law
Therefore
 

F

F

m
a

NET
 p
 F  t
The principle of Conservation of Momentum
 Conserved means doesn’t change

A system cannot change its momentum by itself.
Unless a system is acted on by a NET external force the initial
momentum of a system must equal the final momentum of a
system.
However, two or more systems may exchange momentum.
We will study how these changes occur.
4
Physics 3310-XX
Example Problem:
Water leaves a hose at the rate of 1.5 kg/s with a velocity of 20
m/s, and is aimed at the side of the car, which brings the water to
rest. (We can assume no splash back occurs) What is the force
exerted by the water on the car?
5
Physics 3310-XX
Example Problem:
Conservation of Momentum – Collide & Stick together
A bullet whose mass, m, is 50.0g is fired horizontally with a speed, v, of
1,100 m/s by a rifle whose mass, M = 6.0 kg that is initially at rest. What is
the speed of the rifle when it recoils?
Mossberg
Mossberg
v
vrifle
Example Problem:
Conservation of Momentum – Collide & Stick together
A bullet whose mass, m, is 50.0g is fired horizontally with a speed, v, of
1,100 m/s into a large wooden block of mass, M = 6.0 kg that is initially at
rest on a horizontal table. If the block is free to slide without friction across
the table, what speed will it acquire after it has absorbed a bullet?
Mossberg
M
v
No Friction
6
Physics 3310-XX
Example Problem:
Conservation of momentum – Collide & Stick together
A 10,000 kg railroad car traveling at a speed of 24.0 m/s strikes an
identical railroad car that is at rest. If the cars lock together as a
result of the collision, what is their common speed afterward?
7
Physics 3310-XX
Example Problem:
Conservation of momentum – Internal Motion Problem
A man with a mass of 70 kg is standing on the front end of a flat railroad car,
which has a mass of 1,000 kg and a length of 10 m. The railroad car is
initially at rest relative to the track. The man then walks from one end of the
car t to the other at a speed of 1.0 m/s relative to the track. Assume there is
no friction in the wheels of the railroad car. (a) What happens to the cart
while the man is walking? (b) How long does it take him to reach the other
end of the car? (c) What happens when the man stops at the rear of the car?
1 m/s
+x
8
Physics 3310-XX
1.
(II) A child in a boat throws a 5.40-kg package out horizontally with a speed of 10.0m/s, (Fig. 7-28).
Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the
child is 26.0 kg and that of the boat is 55.0 kg.
2.
(II) Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of
1300 kg/s with a speed of 40,000 m/s (at the moment of takeoff).
9
Physics 3310-XX
3.
(II) A 12,500-kg railroad car travels alone on a level frictionless track with a constant speed of 18.0
m/s. A 5750-kg additional load is dropped into the car. What then will be the car’s speed?
4.
(II) A gun is fired vertically into a 1.40-kg block of wood at rest directly above it. If the bullet has a
mass of 21.0 g and a speed of 210 m/s, how high will the block rise into the air after the bullet becomes
embedded in it?
5.
(II) A 15-g bullet strikes and becomes embedded in a 1.10-kg block of wood placed on a horizontal
surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface
is 0.25, and the impact drives the block a distance of 9.5 m before it comes to rest, what is the muzzle
speed of the bullet?
10
Physics 3310-XX
6.
(II) An Atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction of its
velocity and the new nucleus slows to 350 m/s. If the alpha particle has a mass of 4.0 u and the
original nucleus has a mass of 222 u, what speed does the alpha particle have when it is emitted?
7.
(II) A 13-g bullet traveling 230 m/s penetrates a 2.0-kg block of wood and emerges going 170 m/s. If
the block is stationary on a frictionless surface when hit, how fast does it move after the bullet
emerges?
11
Physics 3310-XX
Definition of Impulse
 p
RECALL:  F  t

  
Therefore: p  p f  pi   FNET t
The impulse of the external forces acting on a system is
equal to the change in the system’s momentum

J  IMPULSE





J  ( FNET )t  p f  pi  p
 tf

J   F (t )dt  p
ti
Area bounded by F(t) function and x-axis
Net External Force vs. Time
10
( Fnet )y
8
6
Area = Change in Momentum
4
2
0
ti
tf
12
Physics 3310-XX
IMPULSE AND COLLISIONS
Collision – When two or more objects come close together or hit
and exert forces on each other for a short time
Impulse Forces – Forces that are exerted for a short time interval
Impulse Force Function
Force vs. Time
F (avg)
ti
t
f
The average Force over the interval t is:

1
Favg 
t

J
F

t

 net
t



J
Favg is the constant force which gives the same impulse, , in the
time interval t.
13
Physics 3310-XX
Example Problem: Impulse
A pitched 140 g baseball, in horizontal flight with a speed vi of 39
m/s, is struck by a batter. After leaving the bat, the ball travels in
the opposite direction
with a speed vf, also 39 m/s.

a. What impulse, J , acts on the ball while it is in contact with the
bat?
b. The impact time, t, for the baseball-bat collision is 1.2 ms, a
typical value. What average force acts on the baseball?
14
Physics 3310-XX
Example Problem: Series of collisions
A machine gun fires R bullets per second. Each bullet has mass, m
and speed, v. The bullets strike a fixed target, where they stop.
What is the average force exerted on the target over a time that is
long compared with the time between bullets?
The graph shows the instantaneous force on the target.
20F
Force of bullet impacts vs. Time
15
10
Favg 5
0
t
15
Physics 3310-XX
8.
(I) A tennis ball may leave the racket of a top player on the serve with a speed of 65.0 m/s. If the ball’s
mass is 0.0600 kg and it is in contact with the racket for 0.0300 s, what is the average force on the
ball? Would this force be large enough to lift a 60-kg person?
9.
(II) A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact
with the ball for 5.0 x 10-3 s. Find (a) the impulse imparted to the golf ball and (b) the average force
exerted on the ball by the golf club.
10. (II) A tennis ball of mass m = 0.060 kg and speed v = 25 m/s strikes a wall at a 45 angle and
rebounds with the same speed at 45 (Fig 7-29). What is the impulse given to the wall?
16
Physics 3310-XX
11. (II) A 115-kg fullback is running at 4.0 m/s to the east and is stopped in 0.75 s by a head-on tackle by a
tackler running due west. Calculate (a) the original momentum of the fullback, (b) the impulse exerted
on the fullback, (c) the impulse exerted on the tackler, and (d) the average force exerted on the tackler.
17
Physics 3310-XX
ELASTIC COLLISIONS
Consider a 2 body system:
m1
v1i
m2
v2i = 0m/s
Assume closed system (no mass enters or leaves)
Assume isolated system (no external forces act on it)

The linear momentum, p , is always conserved in a closed,
isolated system, whether or not the collision is elastic,
because the forces are all internal. (Internal forces cancel
out because they act on the same object or system.)
If the kinetic energy of the system is also conserved then
the collision is elastic.
Therefore, Elastic collisions conserve energy and
momentum.
m1
v1i
INITIAL
If elastic then:
m2
v2i = 0m/s


pi ( system)  p f ( system)
m1
v1f
m2
v2f
FINAL



m1v1i  m1v1 f  m2 v2 f
and
KEi  KE f
1
1
1
m1v12i  m1v12f  m2v22 f
2
2
2
18
Physics 3310-XX
Example Problem:
Elastic Collisions – Equal mass, 90o, after collision problem
Two particles of equal masses have an elastic collision, the target particle
being initially at rest. Show that (unless the collision is head-on) the two
particles will always move off perpendicular to each other after the collision.
+y
v2i = 0m/s
mv1i
+x
m
m
BEFORE
mv1f

mv1f
+y

m
m
mv2f
+x


mv1i
AFTER
mv2f
(a)
(b)
19
Physics 3310-XX
Example Problem:
2-D Collision Problem – Elastic or Inelastic?
A 5 kg particle moves in the +x direction with a velocity of 6 m/s shown in
the figure below. You are told that it makes an elastic collision with a 2 kg
particle that is initially at rest. After the collision, the 2 kg particle moves
with a speed of 5 m/s in the direction 30o above the x-axis, as shown in the
figure below.
What are the x and y components of the 5 kg particle after the collision?
Were you informed correctly? Is this an elastic collision?
5m/s
5kg
6m/s
INITIAL
2kg
2kg
5kg
30o
FINAL
20
Physics 3310-XX
Elastic Collisions: Target ball at rest
Will the incident ball rebound, stop or continue forward?
1
2
v1i
m1
m2
v2i = 0
pi ( system )  p f ( system )
KEi  KE f
m1v1i  m1v1 f  m2 v2 f
m1  v1i  v1 f
m v
m12
 v1i  v1 f
m22
2

2
2f
 v22 f
1
1
1
m1v12i  m1v12f  m2 v22 f
2
2
2
m1  v12i  v12f   m2 v22 f
m1
v1i  v1 f  v1i  v1 f   v22 f

m2
21
Physics 3310-XX
INELASTIC COLLISIONS
Inelastic Collision – a collision in which the kinetic energy of the
system of colliding bodies is not conserved.
EXAMPLE: A ball being dropped to the ground and only
rebounding to ½ its initial height is noticeably inelastic.
 The kinetic energy lost is transformed into some other form of
energy, often thermal.
EXAMPLE: Putty dropped onto a floor does not rebound at all.
 Collisions where there is no rebound at all (the particles stick
together) are called completely inelastic collisions.
NOTE: For completely inelastic collisions momentum is
conserved, as long as the system is isolated and closed.
EXAMPLE: Completely Inelastic Collision
m1
v1i
m2
v2i = 0
BEFORE
(m1+m2)
AFTER
m1
m2
vf
22
Physics 3310-XX
Example Problem:
Inelastic Collision Problem (Ballistic Pendulum)
A ballistic pendulum is a device that was used to measure the speed of
bullets before electronic timing devices were developed. The device
consists of a large block of wood of mass, M = 5.4 kg, hanging from two
long cords. A bullet of mass, m = 9.5 g is fired into the block, coming
quickly to rest. The block + bullet then swing upward, their center of mass
rising a vertical distance, h = 6.3 cm before the pendulum comes
momentarily to rest at the end of its arc.
a) What was the speed of the bullet just prior to the collision?
b) What is the initial kinetic energy of the bullet? How much of this energy
remains as mechanical energy of the swinging pendulum?
m
M
v
h
23
Physics 3310-XX
12. (II) A 0.450-kg ice puck, moving east with a speed of 3.00 m/s, has a head-on collision with a 0.900-kg
puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of
each object after the collision?
13. (II) A pair of bumper cars in an amusement park ride collide elastically
as one approaches the other directly from the rear (Fig. 7-31). One has
a mass of 450 kg and the other 550 kg, owing to differences in
passenger mass. If the lighter one approaches a 4.50 m/s and the other
is moving at 3.70 ms/, calculate (a) their velocities after the collision,
and (b) the change in momentum of each.
24
Physics 3310-XX
14. (III) In a physics lab, a cube slides down a frictionless incline as shown in Fig. 7-32, and elastically
strikes a cube at the bottom that is only one-half its mass. If the incline is 30 cm
high and the table is 90 cm off the floor, where does each cube land?
15. (II) An 18-g rifle bullet traveling 230 m/s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long
string, which makes the pendulum swing upward in an arc. Determine the horizontal component of the
pendulum’s displacement.
25
Physics 3310-XX
16. (II) An explosion breaks an object into two pieces, one of which has 1.5 times the mass of the other. If
7500 J were released in the explosion, how much kinetic energy did each piece acquire?
17. (II) A 1.0 x 103-kg Toyota collides into the rear end of a 2.2 x 10 3-kg Cadillac stopped at a red light.
The bumpers lock, the breaks are locked, and the two cars skid forward 2.8 m before stopping. The
police officer, knowing that the coefficient of kinetic friction between tires and road is 0.40, calculates
the speed of the Toyota at impact. What is that speed?
26
Physics 3310-XX
18. (II) A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The
electron and neutrino are emitted at right angles and have momenta of 9.30 x 10 -23 kg∙m/s, and 5.40 x
10-23 kg∙m/s, respectively. What is the magnitude and direction of the momentum of the second
(recoiling) nucleus?
19. (II) An eagle (m1 = 4.3 kg) moving with speed (v1) = 7.8 m/s is on a collision course with a second
eagle (m2 = 5.6 kg) moving at v2 = 10.2 m/s in the direction at right angles to the first. After they
collide, they hold onto one another. In what direction, and with what speed, are they moving after the
collision?
27
Physics 3310-XX
CENTER OF MASS
The center of mass of a system is the point located by the position


  
vector R , where R is related to the position vectors r1 , r2 , r3 ... of
the particles in the system by the relation.




MR  mr1  mr2  mr3 ...

RCM 
Recall:



mr1  mr2  mr3 ...
M

 r
v
t
AND

 v
a
t
THEREFORE:




MVCM  m1v1  m2v2  m3v3  ...
The momentum of the center of mass is given as
the SUM of the momentum’s of the individual
masses in the system.
AND




MACM  m1a1  m2 a2  m3 a3  ...
F
  F1   F2   F3  ...
F

 MACM
net
net
The Net external force accelerating the center of
mass is the SUM of the external forces
accelerating each of the masses in the system.
28
Physics 3310-XX
EXAMPLE PROBLEM:
UNDERSTANDING CENTER OF MASS
Summing “weighted” position vectors
M = 5.97x1024 kg
0r
10r
m = 7.35x1022 kg
20r
30r
40r
50r
M is located at 1r.
m is located at 61r.
Find the center of mass of these two objects
 Do these masses and distance look familiar?
 Notice the center of mass of this two-body (earth-moon) system is almost
outside of mass M (earth). This is why the earth wobbles in its orbit.
You will learn more about this in the next chapter (rotational motion).
60r
29
Physics 3310-XX
Example Problem:
Y - Position (m)
Center of mass for a discrete number of particles
The system of three particles in the picture below have masses m1
= 5 kg, m2 = 4 kg and m3 = 3 kg. The locations of the particles are
r1 = (1,4) m, r2 = (4,3) m, and r3 = (5,1) m, respectively. Find the
center of mass of this system.
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
m1=5 kg
m2=4 kg
Center of mass (?, ?)
r1
R
r2
m3=3 kg
r3
0
1
2
3
X - Position (m)
4
5
6
30
Physics 3310-XX
LOCATING THE CENTER OF MASS OF AN EXTENDED
OBJECT BY THE SUBTRACTION METHOD
Note: Uniform objects with perfect symmetry will have their
Center of Mass (CM) at the geometric center of the object.
CM
CM
CM
CM
Subtraction Method for determining center of mass
Example Problem:
Find the center of mass of a circular plate of radius 2R from which a disk of
radius R has been removed.
2R
R
Subtraction Method:
Take object 3 to be a system made up of object 1 with positive
density,  (mass/volume) and object 2 with negative density, .
R2 = Area of a Circle
d = thickness of the plate.
-
 = m/v = (m/R2d)
=
m= R2d
m1= R2d
Object 2
Object 1
Object 3
m2= R2d
m1 x1  m2 x2
m1  m2 
Take the origin (0,0) to be at the center of object 1, therefore x1 =0 & x2 = -R
 R 2 d ( R)
1
x3 
 R
2
2
 4R d  R d  3
x3 
31
Physics 3310-XX
EXAMPLE PROBLEM:
Locate the center of mass of the plate that has a circular hole with a
radius of 1 m as shown below.
2m
2m
6m
6m
32
Physics 3310-XX
USES OF THE CENTER OF MASS
 According to  Fnet  MAcm , the center of mass moves as if it were
a particle with the mass of the system.
 C.M. concept useful for solving 2 kinds of problems.
1. Determining the motion of a rigid body by using “simple”
 Fnet  MAcm to understand the motion of the C.M.
2. To Simplify problems involving the motion of 2 or more
particles by a C.M. frame of reference.
33
Physics 3310-XX
20. (I) An empty 1050-kg car has its CM 2.50 m behind the front of the car. How far from the front of the
car will the CM be when two people sit in the front seat 2.80 m from the front of the car, and three
people sit in the back seat 3.90 m from the front? Assume that each person has a mass of 70.0 kg.
21. (II) Three cubes, of side l0, 2l0, and 3l0, are placed next to one another (in contact) with their centers
along a straight line and the l = 2l0 cube in the center (Fig. 7-36). What is the position, along this line
of the CM of this system? Assume the cubes are made of the same uniform material.
34
Physics 3310-XX
22. (III) A uniform circular plate of radius 2R has a circular hole of radius R cut out of it. The center of the
smaller circle is a distance of 0.80R from the center of the larger circle, Fig. 7-38. What is the position
of the center mass of the plate? [Hint: Try subtraction.]
35
Physics 3310-XX
QUESTIONS
1. We claim that momentum is conserved. Yet most moving objects eventually slow down and stop.
Explain.
2.
When a person jumps from a tree to the ground, what happens to the momentum of the person upon
striking the ground?
3.
Why, when you release an inflated but untied balloon, does it fly across the room?
4.
It is said that in ancient times a rich man with a bag of gold coins froze to death stranded on the surface
of a frozen lake. Because the ice was frictionless, he could not push himself to shore. What could he
have done to save himself had he not been so miserly?
36
Physics 3310-XX
5.
How can a rocket change direction when it is far out in space and is essentially in a vacuum?
6.
According to Eq. 5, the longer the impact of time of an impulse, the smaller the force can be for the
same momentum change, and hence the smaller the deformation of the object on which the force acts.
Explain on the basis of the value of “air bags,” which are intended to inflate during an automobile
collision and reduce the possibility of fracture or death.
7.
It used to be common wisdom to build cars to be as rigid as possible to withstand collisions. Today,
though, cars are designed to have “crumple zones” that collapse upon impact. What advantage does
this have?
8.
Why is it easier to hit a home run from a pitched ball that from one tossed in the air by the batter?
37
9.
Physics 3310-XX
Is it possible for a body to receive a larger impulse from a small force than from a large force?
10. A light body and a heavy body have the same kinetic energy. Which has a greater momentum?
11. Is it possible for an object to have momentum without having kinetic energy? Can it have kinetic
energy but no momentum? Explain.
12. At a hydroelectric power plant, water is directed at high speed against turbine blades on an axle that
turns an electric generator. Do you think the turbine blades should be designed so that the water is
brought to a dead stop, or so that the water rebounds?
38
Physics 3310-XX
13. A superball is dropped from a height h onto a hard steel plate (fixed to the Earth), from which it
rebounds at very nearly its original speed. (a) Is the momentum of the ball conserved during any part
of this process? (b) If we consider the ball and Earth as our system, during what parts of the process is
momentum conserved? (c) Answer part (b) for a piece of putty that falls and sticks to the steel plate.
14. Why do you tend to lean backward when carrying a heavy load in your arms?
15. Why is the CM of a 1-m length pipe at its midpoint, whereas this is not true for your arm or leg?
16. Show on a diagram how your CM shifts when you change from a lying position to a sitting position.