Momentum

Conservation of Force

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:

Impulse and Momentum

p = mv

Impulse and Momentum

Because mvf = pf and mvi = 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:

Impulse and Momentum

FΔt = mΔv = pf − pi

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 impulse-angular 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

pCi + pDi = pCf + pDf

0 = pCf + pDf

pCf = −pDf

mCvCf = −mDvDf

After

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:

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

Sum of horizontal components equal to initial momentum

pCi = pCf + pDf

pCf, y + pDf, y = 0

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

Initial angular momentum equal final angular momentum

Earth’s angular momentum constant and conserved. So, length of a day does not change

Lf = Li

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?