Download Chapter 9: Linear Momentum

Chapter 9: Linear Momentum
Topics in Chapter 9:
• the momentum of a particle, and how the net force acting on
a particle causes its momentum to change.
• the circumstances under which the total momentum of a
system of particles is constant (conserved).
• elastic, inelastic, and completely inelastic collisions.
• what’s meant by the center of mass of a system, and what
determines how the center of mass moves.
• how to analyze situations such as rocket propulsion in which
the mass of a body changes as it moves.
Momentum – a new tool for solving problems.
• In many situations, such as a bullet hitting a carrot, we cannot
use Newton’s second law to solve problems because we
know very little about the complicated forces involved.
• In this chapter, we shall introduce momentum and impulse,
and the conservation of momentum, to solve such problems.
Momentum and Newton’s second law
• The momentum of a particle
is the product of its mass and
its velocity:
• Newton’s second law can be
written in terms of momentum:
An isolated system
• Two astronauts push
each other as they
float freely in the
zero-gravity
environment of space.
• There are no external
forces; when this is
the case, we have an
isolated system.
Conservation of momentum
• External forces (the normal
force and gravity) act on the
skaters shown, but their vector
sum is zero.
• Therefore the total momentum
of the skaters is conserved.
• Conservation of momentum:
If the vector sum of the
external forces on a system is
zero, the total momentum of
the system is constant.
Remember that momentum is a vector!
• When applying conservation
of momentum, remember
that momentum is a vector
quantity!
• Use vector addition to add
momenta, as shown at the
right.
Q9.1
A 3.00­kg rifle fires a 0.00500­kg bullet at a speed of 300 m/s. Which force is greater in magnitude: the force that the rifle exerts on the bullet or the force that the bullet exerts on the rifle?
A. The force that the rifle exerts on the bullet is greater.
B. The force that the bullet exerts on the rifle is greater.
C. Both forces have the same magnitude.
D. The answer depends on how the rifle is held.
E. The answer depends on how the rifle is held and on the inner workings of the rifle.
A9.1
A 3.00­kg rifle fires a 0.00500­kg bullet at a speed of 300 m/s. Which force is greater in magnitude: the force that the rifle exerts on the bullet or the force that the bullet exerts on the rifle?
A. The force that the rifle exerts on the bullet is greater.
B. The force that the bullet exerts on the rifle is greater.
C. Both forces have the same magnitude.
D. The answer depends on how the rifle is held.
E. The answer depends on how the rifle is held and on the inner workings of the rifle.
Momentum and Newton’s second law
• The momentum of a particle
is the product of its mass and
its velocity:
• Newton’s second law can be
written in terms of momentum:
Collisions
• A collision is a brief, intense interaction between objects.
• The collision time is short compared with the
timescale of the objects' overall motion.
• Internal forces of the collision are so large that we can
neglect any external forces acting on the system
during the brief collision time.
• Therefore linear momentum is essentially conserved
during collisions.
Elastic and Inelastic Collisions
• In an elastic collision kinetic energy is conserved.
• Therefore, the internal forces in an elastic collision
must be conservative.
• In an inelastic collision, the forces are not conservative
and mechanical energy is lost.
• In a totally inelastic collision, the colliding objects
stick together to form a single composite object.
• Even if a collision is totally inelastic, that doesn't
necessarily mean that all kinetic energy is lost
(momentum must still be conserved).
Elastic collisions: Before
• In an elastic collision, the total kinetic energy of the system is
the same after the collision as before.
Elastic collisions: During
• In an elastic collision, the total kinetic energy of the system is
the same after the collision as before.
Elastic collisions: After
• In an elastic collision, the total kinetic energy of the system is
the same after the collision as before.
Elastic collisions
• Billiard balls deform very little when they collide, and they
quickly spring back from any deformation they do undergo.
• Hence the force of
interaction between the balls
is almost perfectly
conservative, and the
collision is almost perfectly
elastic.
Elastic collisions in one dimension
• Let’s look at a one-dimensional elastic collision between two bodies
A and B, in which all the velocities lie along the same line.
• We will concentrate on the particular case in which body B is at rest
before the collision.
• This type of collision conserves both kinetic energy and momentum.
Elastic collisions in one dimension
• Let’s look at a one-dimensional elastic collision between two
bodies A and B, in which all the velocities lie along the same line.
• We will concentrate on the particular case in which body B is at
rest before the collision.
• Conservation of kinetic energy and momentum give the result for
the final velocities of A and B:
Elastic collisions in one dimension
• When B is much more massive than A, then A reverses its
velocity direction, and B hardly moves.
Elastic collisions in one dimension
• When B is much more massive than A, then A reverses its
velocity direction, and B hardly moves.
Elastic collisions in one dimension
• When A and B have similar masses, then A stops after the
collision and B moves with the original speed of A.
Completely inelastic collisions: Before
• In an inelastic collision, the total kinetic energy after the
collision is less than before the collision.
Completely inelastic collisions: During
• In an inelastic collision, the total kinetic energy after the
collision is less than before the collision.
Completely inelastic collisions: After
• In an inelastic collision, the total kinetic energy after the
collision is less than before the collision.
Inelastic collisions
• Cars are designed so that collisions are inelastic—the
structure of the car absorbs as much of the energy of the
collision as possible.
• This absorbed energy cannot
be recovered, since it goes
into a permanent
deformation of the car.
Inelastic Collisions
Example: Ballistic pendulum.
The ballistic pendulum is a device
used to measure the speed of a
projectile, such as a bullet. The
projectile, of mass m, is fired into a
large block of mass M, which is
suspended like a pendulum. As a
result of the collision, the pendulum
and projectile together swing up to a
maximum height h. Determine the
relationship between the initial
horizontal speed of the projectile, v,
and the maximum height h.
Impulse
• The impulse of a force is the product of the force and the
time interval during which it acts:
• On a graph of
versus time, the impulse is equal to the
area under the curve:
Impulse
Impulse
Impulse and momentum
• Impulse–momentum theorem: The change in momentum of
a particle during a time interval is equal to the impulse of the
net force acting on the particle during that interval:
Impulse and momentum
• When you land after jumping
upward, your momentum
changes from a downward
value to zero.
• It’s best to land with your
knees bent so that your legs
can flex.
• You then take a relatively long time to stop, and the force that
the ground exerts on your legs is small.
• If you land with your legs extended, you stop in a short time,
the force on your legs is larger, and the possibility of injury is
greater.
Center of mass
• We can restate the principle of conservation of momentum in
a useful way by using the concept of center of mass.
• Suppose we have several particles with masses m1, m2, and so
on.
• We define the center of mass of the system as the point at the
position given by:
Center of Mass
For two particles, the center of mass lies closer to the
one with the most mass:
where M is the total mass.
Center of mass of symmetrical objects
Center of mass of symmetrical objects
Motion of the center of mass
• The total momentum of a system is
equal to the total mass times the velocity
of the center of mass.
• The center of mass of the wrench at the
right moves as though all the mass were
concentrated there.
Motion of the Center of Mass
• The center of mass obeys Newton's second law:


Fnet external  M acm
• The parts of a rotating projectile undergo complex
motions, but its center of mass follows a parabolic
arc.
Center of Mass and Translational Motion
The total momentum of a system of particles is equal to
the product of the total mass and the velocity of the
center of mass.
The sum of all the forces acting on a system is equal to
the total mass of the system multiplied by the
acceleration of the center of mass:
Therefore, the center of mass of a system of particles
(or objects) with total mass M moves like a single
particle of mass M acted upon by the same net
external force.
Center of Mass and Translational Motion
• In the absence of any external forces on a system, the center of
mass motion remains unchanged; if it's at rest, it remains in
the same place—no matter what internal forces may act.
• Example: Jumbo, a 4.8-t elephant, walks 19 m toward one end
of the car, but the CM of the 15-t rail car plus elephant doesn't
move. This allows us to find the car's final position (we set the
initial position of the car's CM at the origin):
Question1: Conservation of Momentum
Falling on or off a sled.
An empty sled is sliding on frictionless ice when
Susan drops vertically from a tree above onto the sled.
When she lands, does the sled:
A) speed up
B) slow down, or
C) keep the same speed?
Question1: Conservation of Momentum
Falling on or off a sled.
An empty sled is sliding on frictionless ice when
Susan drops vertically from a tree above onto the sled.
When she lands, does the sled:
A) speed up
B) slow down, or
C) keep the same speed?
Question 2: When a cannonball is fired from a cannon,
why does the cannon recoil backward?
a) The total energy of the cannonball and cannon is
conserved.
b) The energy of the cannon is greater than the energy of
the cannonball.
c) The momentum of the cannon is conserved.
d) The momentum of the cannon is greater than the
momentum of the cannonball.
e) The total momentum of the cannonball and
cannon is conserved.
Question 2: When a cannonball is fired from a cannon,
why does the cannon recoil backward?
a) The total energy of the cannonball and cannon is
conserved.
b) The energy of the cannon is greater than the energy of
the cannonball.
c) The momentum of the cannon is conserved.
d) The momentum of the cannon is greater than the
momentum of the cannonball.
e) The total momentum of the cannonball and cannon
is conserved.
Question 3
Two objects with different masses collide with and stick to each other. Compared to before the collision, the system of two objects after the collision has
A
B
A. the same amount of total momentum and the same total kinetic energy.
B. the same amount of total momentum but less total kinetic energy.
C. less total momentum but the same amount of total kinetic energy.
D. less total momentum and less total kinetic energy.
E. Not enough information is given to decide.
Question 3
Two objects with different masses collide with and stick to each other. Compared to before the collision, the system of two objects after the collision has
A
B
A. the same amount of total momentum and the same total kinetic energy.
B. the same amount of total momentum but less total kinetic energy.
C. less total momentum but the same amount of total kinetic energy.
D. less total momentum and less total kinetic energy.
E. Not enough information is given to decide.
Summary of Chapter 9
• Momentum of an object:
• Newton’s second law:
• Total momentum of an isolated system of objects is conserved.
• During a collision, the colliding objects can be considered to be
an isolated system even if external forces exist, as long as they
are not too large.
• Momentum will therefore be conserved during collisions.
Summary of Chapter 9, cont.
• Impulse:
• In an elastic collision, total kinetic energy is also
conserved.
• In an inelastic collision, some kinetic energy is lost.
• In a completely inelastic collision, the two objects stick
together after the collision.
• The center of mass of a system is the point at which
external forces can be considered to act.