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Chapter 12: Momentum. 12.1 Momentum 12.2 Force is the Rate of Change of Momentum 12.3 Angular Momentum. Chapter 12 Objectives. Calculate the linear momentum of a moving object given the mass and velocity. Describe the relationship between linear momentum and force.

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Chapter 12: Momentum

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### Chapter 12: Momentum

12.1 Momentum

12.2 Force is the Rate of Change of Momentum

12.3 Angular Momentum

### Chapter 12 Objectives

Calculate the linear momentum of a moving object given the mass and velocity.

Describe the relationship between linear momentum and force.

Solve a one-dimensional elastic collision problem using momentum conservation.

Describe the properties of angular momentum in a system—for instance, a bicycle.

Calculate the angular momentum of a rotating object with a simple shape.

### Chapter Vocabulary

• angular momentum

• collision

• law of conservation of

• momentum

• elastic collision

• gyroscope

• impulse

• inelastic collision

• linear momentum

• momentum

### Inv 12.1 Momentum

Investigation Key Question:

What are some useful properties of momentum?

### 12.1 Momentum

• Momentum is a property of moving matter.

• Momentum describes the tendency of objects to keep going in the same direction with the same speed.

• Changes in momentum result from forces or create forces.

### 12.1 Momentum

• The momentum of a ball depends on its mass and velocity.

• Ball B has more momentum than ball A.

### 12.1 Momentum and Inertia

• Inertia is another property of mass that resists changes in velocity; however, inertia depends only on mass.

• Inertia is a scalar quantity.

• Momentum is a property of moving mass that resists changes in a moving object’s velocity.

• Momentum is a vector quantity.

### 12.1 Momentum

• Ball A is 1 kg moving 1m/sec, ball B is 1kg at 3 m/sec.

• A 1 N force is applied to deflect the motion of each ball.

• What happens?

• Does the force deflect both balls equally?

• Ball B deflects much less than ball A when the same force is applied because ball B had a greater initial momentum.

### 12.1 Kinetic Energy and Momentum

• Kinetic energy and momentum are different quantities, even though both depend on mass and speed.

• Kinetic energy is a scalar quantity.

• Momentum is a vector, so it always depends on direction.

Two balls with the same mass and speed have the same kinetic energy but opposite momentum.

p = m v

### 12.1 Calculating Momentum

• The momentum of a moving object is its mass multiplied by its velocity.

• That means momentum increases with both mass and velocity.

Momentum

(kg m/sec)

Velocity (m/sec)

Mass (kg)

### Comparing momentum

A car is traveling at a velocity of 13.5 m/sec (30 mph) north on a straight road. The mass of the car is 1,300 kg. A motorcycle passes the car at a speed of 30 m/sec (67 mph). The motorcycle (with rider) has a mass of 350 kg. Calculate and compare the momentum of the car and motorcycle.

• You are asked for momentum.

• You are given masses and velocities.

• Use: p = m v

• Solve for car: p = (1,300 kg) (13.5 m/s) = 17,550 kg m/s

• Solve for cycle: p = (350 kg) (30 m/s) = 10,500 kg m/s

• The car has more momentum even though it is going much slower.

### 12.1 Conservation of Momentum

• The law of conservation of momentum states when a system of interacting objects is not influenced by outside forces (like friction), the total momentum of the system cannot change.

If you throw a rock forward from a skateboard, you will move backward in response.

### 12.1 Collisions in One Dimension

• A collisionoccurs when two or more objects hit each other.

• During a collision, momentum is transferred from one object to another.

• Collisions can be elastic

orinelastic.

### Elastic collisions

Two 0.165 kg billiard balls roll toward each other and collide head-on.

Initially, the 5-ball has a velocity of 0.5 m/s.

The 10-ball has an initial velocity of -0.7 m/s.

The collision is elastic and the 10-ball rebounds with a velocity of 0.4 m/s, reversing its direction.

What is the velocity of the 5-ball after the collision?

### Elastic collisions

• You are asked for 10-ball’s velocity after collision.

• You are given mass, initial velocities, 5-ball’s final velocity.

• Diagram the motion, use m1v1 + m2v2 = m1v3 + m2v4

• Solve for V3 : (0.165 kg)(0.5 m/s) + (0.165 kg) (-0.7 kg)=

(0.165 kg) v3 + (0.165 kg) (0.4 m/s)

• V3 = -0.6 m/s

### Inelastic collisions

A train car moving to the right at 10 m/s collides with a parked train car.

They stick together and roll along the track.

If the moving car has a mass of 8,000 kg and the parked car has a mass of 2,000 kg, what is their combined velocity after the collision?

• You are asked for the final velocity.

• You are given masses, and initial velocity of moving train car.

### Inelastic collisions

• Diagram the problem, use m1v1 + m2v2 = (m1v1 +m2v2) v3

• Solve for v3= (8,000 kg)(10 m/s) + (2,000 kg)(0 m/s)

(8,000 + 2,000 kg)

v3= 8 m/s

The train cars moving together to right at 8 m/s.

### 12.1 Collisions in 2 and 3 Dimensions

• Most real-life collisions do not occur in one dimension.

• In a two or three-dimensional collision, objects move at angles to each other before or after they collide.

• In order to analyze two-dimensional collisions you need to look at each dimension separately.

• Momentum is conserved separatelyin the xand ydirections.

12.1 Collisions in 2 and 3 Dimensions

### Chapter 12: Momentum

12.1 Momentum

12.2 Force is the Rate of Change of Momentum

12.3 Angular Momentum

Investigation Key Question:

How are force and momentum related?

### 12.2 Force is the Rate of Change of Momentum

• Momentum changes when a net force is applied.

• The inverse is also true:

• If momentum changes, forces are created.

• If momentum changes quickly, large forces are involved.

F = D p

D t

### 12.2 Force and Momentum Change

The relationship between force and motion follows directly from Newton's second law.

Force (N)

Change in momentum

(kg m/sec)

Change in time (sec)

### Calculating force

• You are asked for force exerted on rocket.

• You are given rate of fuel ejection and speed of rocket

• Use F = Δ ÷Δt

• Solve: Δ = (100 kg) (-25,000 kg m/s) ÷ (1s) = - 25,000 N

• The fuel exerts and equal and opposite force on rocket of +25,000 N.

Starting at rest, an 1,800 kg rocket takes off, ejecting 100 kg of fuel per second out of its nozzle at a speed of 2,500 m/sec. Calculate the force on the rocket from the change in momentum of the fuel.

### 12.2 Impulse

• The product of a force and the time the force acts is called the impulse.

• Impulse is a way to measure a change in momentum because it is not always possible to calculate force and time individually since collisions happen so fast.

F D t = D p

### 12.2 Force and Momentum Change

To find the impulse, you rearrange the momentum form of the second law.

Impulse (N•sec)

Change in

momentum

(kg•m/sec)

Impulse can be expressed in kg•m/sec (momentum units) or in N•sec.

### Chapter 12: Momentum

12.1 Momentum

12.2 Force is the Rate of Change of Momentum

12.3 Angular Momentum

Investigation Key Question:

How does the first law apply to rotational motion?

### 12.3 Angular Momentum

• Momentum resulting from an object moving in linear motion is called linear momentum.

• Momentum resulting from the rotation (or spin) of an object is called angular momentum.

### 12.3 Conservation of Angular Momentum

• Angular momentum is important because it obeys a conservation law, as does linear momentum.

• The total angular momentum of a closed system stays the same.

L = Iw

### 12.3 Calculating angular momentum

Angular momentum is calculated in a similar way to linear momentum, except the mass and velocity are replaced by the moment of inertia and angular velocity.

Moment of inertia

(kg m2)

Angular

momentum

(kg m/sec2)

Angular

velocity

### 12.3 Calculating angular momentum

• The moment of inertia of an object is the average of mass times radius squared for the whole object.

• Since the radius is measured from the axis of rotation, the moment of inertia depends on the axis of rotation.

### Calculating angular momentum

An artist is making a moving metal sculpture. She takes two identical 1 kg metal bars and bends one into a hoop with a radius of 0.16 m. The hoop spins like a wheel. The other bar is left straight with a length of 1 meter. The straight bar spins around its center. Both have an angular velocity of 1 rad/sec. Calculate the angular momentum of each and decide which would be harder to stop.

• You are asked for angular momentum.

• You are given mass, shape, and angular velocity.

• Hint: both rotate about y axis.

• Use L= I, Ihoop = mr2, Ibar = 1/12 ml2

### Calculating angular momentum

• Solve hoop: Ihoop= (1 kg) (0.16 m)2 = 0.026 kg m2

• Lhoop= (1 rad/s) (0.026 kg m2) = 0.026 kg m2/s

• Solve bar: Ibar= (1/12)(1 kg) (1 m)2 = 0.083 kg m2

• Lbar = (1 rad/s) (0.083 kg m2) = 0.083 kg m2/s

• The bar has more than 3x the angular momentum of the hoop, so it is harder to stop.

### 12.3 Gyroscopes angular momentum

• A gyroscope is a device that contains a spinning object with a lot of angular momentum.

• Gyroscopes can do amazing tricks because they conserve angular momentum.

• For example, a spinning gyroscope can easily balance on a pencil point.

### 12.3 Gyroscopes angular momentum

• A gyroscope on the space shuttle is mounted at the center of mass, allowing a computer to measure rotation of the spacecraft in three dimensions.

• An on-board computer is able to accurately measure the rotation of the shuttle and maintain its orientation in space.

Jet Engines

• Nearly all modern airplanes use jet propulsion to fly. Jet engines and rockets work because of conservation of linear momentum.

• A rocket engine uses the same principles as a jet, except that in space, there is no oxygen.

• Most rockets have to carry so much oxygen and fuel that the payload of people or satellites is usually less than 5 percent of the total mass of the rocket at launch.