Angular Momentum

Definition of Angular Momentum First – definition of torque: τ = Frsinθ the direction is either clockwise or counterclockwise a net torque produces a change in angular motion F

Angular Momentum Angular momentum (L) – how difficult to stop an object from rotating. Product of an objects moment of inertia (I) and its angular velocity (w). If the net torque on an object is zero then angular momentum is conserved. L=mvr sinθ Derive L for a point particle moving in a circle

Example Rank the following from largest to smallest angular momentum.

answer

Review Important Ideas Angular Velocity = Angular Displacement/time Just as it takes a impulse (force applied over a time, FΔt) to change the momentum an object, it takes an angular impulse (torque applied over a time, τ Δt ) to change an object's angular momentum.

The Conservation of Angular Momentum If no net external torques act on a system then the system’s angular momentum, l, remains constant. Speed and radius can change just as long as angular momentum is constant.

Example The figure below shows two masses held together by a thread on a rod that is rotating about its center with angular velocity, ω. If the thread breaks, what happens to the system's (a) angular momentum and (b) angular speed. (Increase, decrease or remains the same)

Examples of the Conservation of Angular Momentum Natalia Kanounnikova World Record Spin RPM

Diving 18 meter high dive Amazing Fouette Turns on Pointe Playground Physics

Example Four identical masses rotate about a common axis, 1.2 meters from the center. Each mass is 2.5 kg, and the system rotates at 2  rad/sec. The rods connecting them are “lightweight”. Find the total angular momentum of this system. 1.2 m 2.5 kg

Answer 90.5 kg m 2 /s

Example Continued The four masses are then pulled toward the center until their radii are 0.5 meters. This is done in such a manner that no external torque acts on the system. What is the new angular speed of the system?

36.2 rad/s

Conservation of Angular Momentum

Warm up

Rotational Inertia & Kinetic Energy

Translation vs Rotational equations Displacement Velocity Acceleration Time Force/Torque Momentum Kinetic Energy

Rotational Kinetic Energy Objects traveling with a translational velocity have energy of motion, known as translation KE = 1/2 mv 2 Objects traveling with angular (rotational) velocity have rotational KE = 1/2 Iω 2

KE of a basketball A.62kg basketball flies through the air with a velocity of 8.0 m/s. What is the translational KE of the basketball? The same basketball with a radius of 0.38m also spins about its axis with an angular velocity of 5 rad/sec. Determine its moment of inertia and its rotational KE. What is the total KE of the basketball?

A 1.20 kg solid cylinder disk with a radius of 10.0 cm rolls without slipping. The linear speed of the disk is v = 1.41 m/s. (a) Find the translational kinetic energy. (b) Find the rotational kinetic energy. (c) Find the total kinetic energy.

Solid Sphere Rolling Down an Incline

Finding the velocity of Solid Sphere Rolling Down an Incline Conservation of energy I = 2/5 MR 2 Now you try for a solid cylinder and hollow cylinder

Who wins the race? A sphere, a cylinder, and a hoop, all of mass M and radius R, are released from rest and roll down a ramp of height h and slope θ. They are joined by a particle of mass M that slides down the ramp without friction. Who wins? Who loses?

A bowling ball A bowling ball that has an 11 cm radius and a 7.2 kg mass is rolling without slipping at 2.0 m/s on a horizontal ball return. It continues to roll without slipping up a hill to a height h before momentarily coming to rest and then rolling back down the hill. How high is the hill?

A skater begins a spin with her arms out. Her rotational inertia is 5 kg m 2, and her angular velocity is 1.2 revs/second. She draws her arms in to decrease her rotational inertia to 2 kgm 2, which increases her angular speed. –What is her angular speed (in radians/second) after drawing her arms in? –What is her kinetic energy both at the beginning and at the end of her spin? –How much work does she do pulling in her arms?

A 0.5 kg mass is attached to the end of a 0.2 kg meter stick, the rotational inertia of the meter stick is kgm 2. The meter stick starts from rest in the position shown, and is allowed to swing downward. a)What is the rotational inertia of the mass, relative to the pivot? b)What is the rotational inertia of the mass/meterstick system? c) What is the potential energy lost by the mass? d) What is the potential energy lost by the meter stick? e) What is the angular velocity of the system when the mass reaches its lowest point? f) What is the velocity of the mass at its lowest point?

A wheel with a mass of 3 kg and a radius of 0.2 meters is attached to a 0.4 kg mass by a string, as shown. The mass is dropped 2 meters, from rest –What is the rotational inertia of the wheel? –What is the change in PE for the mass? –What is the velocity of the mass after it has fallen the 2 meters?

A wheel with a rotational inertia of 0.2 kgm 2 is rotating at 48 radians/second, when a brake is applied as shown. The brake applies a force of friction of 8N to the edge of the wheel. a.What is the angular acceleration of the wheel? b.How long will it take the wheel to come to a complete stop? c.Through what angle will the wheel rotate? d.How far will a point on the edge of the wheel travel?