Inv 12.3 Angular Momentum Investigation Key Question: How does the first law apply to rotational motion?

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. 2

Chapter Vocabulary angular momentum collision law of conservation of elastic collision gyroscope impulse inelastic collision linear momentum

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.