Chapter 12: Momentum 12.1 Momentum 12.2 Force is the Rate of Change of Momentum 12.3 Angular Momentum 2

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. 3

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? 5

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 Conservation of Momentum To see the relationship, consider two balls connected by a spring. The balls are motionless and therefore have no momentum. When you compress the spring, the third law says the balls exert equal forces (through the springs) in opposite directions on one another, -F1 = F2 Remember from Newton’s second law that the equal and opposite forces create opposite accelerations, which create opposite velocities. The accelerations are inversely proportional to the masses, so the velocities are also inversely proportional to the masses. Heavy objects end up with less velocity and light objects with more velocity. The velocities caused by the original equal and opposite forces are exactly as predicted by the law of momentum conservation.

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.

12.1 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 separately in the x and y directions.

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 23

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 = D p D t Change in momentum (kg m/sec) Change in time (sec)

Calculating force You are asked for force exerted on rocket. 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. 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.

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 D t = D p 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 30

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 w Angular velocity (rad/sec)

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.