Download p - Effingham County Schools

Momentum
Conservation of Force
Impulse and Momentum

Impulse and Momentum
Impulse and Momentum

Impulse and Momentum

The right side of the equation FΔt = mΔv, mΔv involves the change in
velocity: Δv = vf − vi

Therefore, mΔv = mvf − mvi

The product of the object’s mass, m, and the object’s velocity, v, is
defined as the momentum of the object. Momentum is measured in
kg·m/s. An object’s momentum, also known as linear momentum, is
represented by the following equation:
p = mv
Impulse and Momentum

Impulse and Momentum

Because mvf = pf and mvi = pi:
FΔt = mΔv = pf − pi

pf − pi, describes change in momentum of an object

The right side of this equation, pf − pi, describes the change in
momentum of an object. Thus, the impulse on an object is equal to the
change in its momentum, which is called the impulse-momentum
theorem

The impulse-momentum theorem is represented by the following
equation:
FΔt = pf − pi
Impulse and Momentum

Impulse and Momentum

If the force on an object is constant, the impulse is the product of the
force multiplied by the time interval over which it acts

Because velocity is a vector, momentum also is a vector

Similarly, impulse is a vector because force is a vector

This means that signs will be important for motion in one dimension
Impulse and Momentum

Impulse-Momentum Theorem

Look at the change in momentum of a baseball. The impulse, that is
the area under the curve, is approximately 13.1 N·s. The direction of
the impulse is in the direction of the force. Therefore, the change in
momentum of the ball is also 13.1 N·s
Impulse and Momentum

Impulse-Momentum Theorem

What is the final momentum of ball after collision if initial momentum
is -5.5 kg  m/s?

What is the baseball’s final velocity?
Impulse and Momentum

Angular Momentum

The angular velocity of a rotating object changes only if torque is
applied to it

This is a statement of Newton’s law for rotating motion, τ = IΔω/Δt

This equation can be rearranged in the same way as Newton’s second
law of motion was, to produce τΔt = IΔω

The left side of this equation is the angular impulse of the rotating
object and the right side can be rewritten as Δω = ωf− ωi
Impulse and Momentum

Angular Momentum

The angular momentum of an object is equal to the product of a
rotating object’s moment of inertia and angular velocity. Angular
momentum is measured in kg·m2/s
L = Iω
Impulse and Momentum

Angular Momentum

Just as the linear momentum of an object changes when an impulse
acts on it, the angular momentum of an object changes when an
angular impulse acts on it

Thus, the angular impulse on the object is equal to the change in the
object’s angular momentum, which is called the angular impulseangular momentum theorem

The angular impulse-angular momentum theorem is represented by the
following equation:
τΔt = Lf − Li
Impulse and Momentum

Angular Momentum

If there are no forces acting on an object, its linear momentum is
constant

If there are no torques acting on an object, its angular momentum is
also constant

Because an object’s mass cannot be changed, if its momentum is
constant, then its velocity is also constant
Impulse and Momentum

Angular Momentum

In the case of angular momentum, however, the object’s angular
velocity does not remain constant.

This is because the moment of inertia
depends on the object’s mass and the
way it is distributed about the axis of
rotation or revolution

Thus, the angular velocity of an object
can change even if no torques are acting
on it
Impulse and Momentum

Angular Momentum

The diver uses the diving board to apply an external torque to her body

Then, the diver moves her center of mass in front of her feet and uses
the board to give a final upward push to her feet

This torque acts over time, Δt, and thus increases the angular
momentum of the diver

Before the diver reaches the water,
she can change her angular velocity
by changing her moment of inertia.
She may go into a tuck position,
grabbing her knees with her hands
Impulse and Momentum

Angular Momentum

By moving her mass closer to the axis of rotation, the diver decreases
her moment of inertia and increases her angular velocity

When she nears the water, she stretches her
body straight, thereby increasing the moment
of inertia and reducing the angular velocity

As a result, she goes straight into the water
Impulse and Momentum

Angular Momentum
Impulse and Momentum

Impulse and Momentum

A golfer uses a club to hit a 45 g golf ball resting on an elevated tee, so
that the golf ball leaves the tee at a horizontal speed of 38 m/s.

What is the impulse on the golf ball?

What is the average force that the club exerts on the golf ball if they
are in contact for 2.0 x 10-3 s?

What average force does the golf ball exert on the club during this
time interval?
Impulse and Momentum

Impulse and Momentum

A single uranium atom has a mass of 3.97 x 10-25 kg. It decays into the
nucleus of a thorium atom by emitting an alpha particle at a speed of
2.10 x 107 m/s. The mass of an alpha particle is 6.68 x 10-27 kg. What
is the recoil speed of the thorium nucleus?

Two cars enter an icy intersection. Car 1, with a mass of 2.50 x 103 kg,
is heading east at 20.0 m/s, and car 2, with a mass of 1.45 x 103 kg is
going north at 30.0 m/s. The two vehicles collide and stick together.
What is the speed and direction of the cars as they skid away together
just after colliding?
Conservation of Momentum

Two-Particle Collisions
Conservation of Momentum

Momentum in a Closed, Isolated System

Under what conditions is momentum of system of two balls
conserved?

First condition: no balls lost and no balls gained. Closed system:
one which does not gain or lose mass

Second condition: forces internal (no forces acting on system by
objects outside it)

When net external force on closed system zero, system is an
isolated system
Impulse and Momentum

Momentum in a Closed, Isolated System

No system on Earth absolutely isolated, because always some
interactions between system and its surroundings

Often interactions small enough to be ignored

Systems contain any number of objects, and objects stick together
or come apart in collision

Law of conservation of momentum: momentum of any closed,
isolated system does not change
Conservation of Momentum

Momentum in a Closed, Isolated System

A 1875-kg car going 23 m/s rear-ends a 1025-kg compact car going
17 m/s on ice in the same direction. The two cars stick together.
How fast do the two cars move together immediately after the
collision?
Conservation of Momentum

Recoil

The momentum of a baseball changes when the external force of a
bat is exerted on it. The baseball, therefore, is not an isolated
system

On the other hand, the total momentum of two colliding balls
within an isolated system does not change because all forces are
between the objects within the system
Conservation of Momentum

Recoil

Assume that a girl and a boy are skating on a smooth surface with
no external forces. They both start at rest, one behind the other.
Skater C, the boy, gives skater D, the girl, a push. Find the final
velocities of the two in-line skaters
Conservation of Momentum

Recoil

After clashing with each other, both skaters are moving, making
this situation similar to that of an explosion. Because the push was
an internal force, you can use the law of conservation of
momentum to find the skaters’ relative velocities


The total momentum of the system was zero before the push.
Therefore, it must be zero after the push
Conservation of Momentum

Recoil
Before
After
pCi + pDi
=
pCf + pDf
0
=
pCf + pDf
pCf
=
−pDf
mCvCf
=
−mDvDf
Conservation of Momentum

Recoil

Are the skaters’ velocities equal and opposite?

The last equation, for the velocity of skater C, can be rewritten as
follows:

Velocities depend on skaters’ relative masses. Less massive skater
moves at greater velocity
Conservation of Momentum

Propulsion in Space

How does a rocket in space change its velocity?

Rocket carries both fuel and oxidizer. When the fuel and oxidizer
combine in rocket motor, resulting hot gases leave exhaust nozzle
at high speed

Rocket and chemicals are closed system

Forces that expel gases internal forces, so system also isolated

Thus, objects in space accelerate using law of conservation of
momentum and Newton’s third law of motion
Conservation of Momentum

Propulsion in Space

Deep Space 1 performed flyby of asteroid Braille in 1999

Had ion engine that exerted as much force as sheet of paper resting
on person’s hand

In ion engine, xenon atoms expelled at speed of 30 km/s, produced
force of only 0.092 N. Runs continuously for days, weeks, or
months

Impulse delivered large enough to increase momentum
Conservation of Momentum

Two-Dimensional Collisions

Two billiard balls system

Original momentum of moving ball pCi and momentum of the
stationary ball zero

Momentum of system before collision
equal to pCi

After collision, both billiard balls
moving and have momenta

If friction ignored, system closed and
isolated
Conservation of Momentum

Two-Dimensional Collisions

Law of conservation of momentum used

Initial momentum equals vector sum of final momenta. So:
pCi = pCf + pDf

Components of vectors before and after collision equal. X-axis in
direction of initial momentum. Y-component of initial momentum
zero

Sum of final y-components also zero
pCf, y + pDf, y = 0

Sum of horizontal components equal to initial momentum
pCi = pCf, x + pDf, x
Conservation of Momentum

Conservation of Angular Momentum

Like linear momentum, angular momentum can be conserved

The law of conservation of angular momentum states that if no net
external torque acts on an object, then its angular momentum does
not change
Lf = Li

Initial angular momentum equal final angular momentum

Earth’s angular momentum constant and conserved. So, length of a
day does not change
Conservation of Momentum

Conservation of Angular Momentum

When ice skater pulls in arms, he begins spinning faster

Without external torque, angular momentum does not change; L =
Iω constant

Increased angular velocity makes decreased moment of inertia

By pulling arms close to body, ice-skater brings more mass closer
to axis of rotation, decreasing radius of rotation and decreasing his
moment of inertia
Li = Lf
so, Iiωi = Ifωf
Conservation of Momentum

Conservation of Angular Momentum

Frequency is f = ω/2π, so:
Conservation of Momentum

Conservation of Angular Momentum

If torque-free object starts with no angular momentum, must
continue with no angular momentum

Thus, if part of an object rotates in one direction, another part must
rotate in the opposite direction

For example, if you switch on a loosely held electric drill, the drill
body will rotate in the direction opposite to the rotation of the
motor and bit
Conservation of Momentum

Tops and Gyroscopes

Because of conservation of angular momentum, direction of
rotation of a spinning object can be changed only by applying a
torque

When a top is vertical, there is no
torque on it, and the direction of
its rotation does not change

If the top is tipped a torque tries
to rotate it downward. Rather than
tipping over, however, the upper end
of the top revolves, or precesses
slowly about the vertical axis
Conservation of Momentum

Tops and Gyroscopes

Gyroscope – wheel or disk that spins rapidly around one axis while
being free to rotate around one or two other axes

Direction of large angular momentum changed only by applying
appropriate torque

Without torque, direction of axis
of rotation does not change
Conservation of Momentum

Tops and Gyroscopes

Gyroscopes are used in airplanes, submarines, and spacecraft to keep
an unchanging reference direction

Giant gyroscopes are used in cruise ships to reduce their motion in
rough water. Gyroscopic compasses, unlike magnetic compasses,
maintain direction even when they are not on a level surface
Conservation of Momentum

Momentum

At 9.0 s after takeoff, a 250 kg rocket attains a vertical velocity of 120
m/s.

What is the impulse on the rocket?

What is the average force on the rocket?

What is its altitude?
Conservation of Momentum

Momentum

In a circus act, a 18 kg dog is trained to jump onto a 3.0 kg skateboard
moving with a velocity of 0.14 m/s. At what velocity does the dog
jump onto the skateboard if afterward the velocity of the dog and
skateboard is -10.0 m/s?