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 elastic collision gyroscope impulse inelastic collision linear 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.

12.1 Calculating Momentum p = m v 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) p = m v Velocity (m/sec) Mass (kg)

Comparing momentum You are asked for 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 collision occurs when two or more objects hit each other. During a collision, momentum is transferred from one object to another. Collisions can be elastic or inelastic.

12.1 Collisions

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 You are asked for the final velocity. 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.

12.2 Force is the Rate of Change of 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.

12.2 Force and Momentum Change The relationship between force and motion follows directly from Newton's second law. Force (N) F = Δ p Δ t Change in momentum (kg m/sec) Change in time (sec)

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.

12.2 Force and Momentum Change To find the impulse, you rearrange the momentum form of the second law. Impulse (N•sec) F Δ t = Δ p Change in momentum (kg•m/sec) Impulse can be expressed in kg•m/sec (momentum units) or in N•sec.

Inv 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.

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) L = I ω Angular velocity (rad/sec)

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.