Angular Momentum and Two-Dimensional Rotational

Dynamics

8.01 W9D2

Next Reading Assignment: W09D3

Quiz 7: Momentum

Review Previous Table Problem

Review Concept Question:

Table Problem:

Table Problem

Table Problem

Table Problem

Concept Question

Concept Question

Concept Question: Torque

Consider two vectors with x > 0 and with Fx > 0 and Fz > 0 . The cross product points in the

1)  + x-direction 2)  -x-direction 3)  +y-direction 4)  -y-direction 5)  +z-direction 6)  -z-direction 7)  None of the above directions

F = Fx̂i + Fzk̂

Concept Question: Change in Angular Momentum

A person spins a tennis ball on a string in a horizontal circle with velocity (so that the axis of rotation is vertical). At the point indicated below, the ball is given a sharp blow (force ) in the forward direction. This causes a change in angular momentum in the

1.  direction

2.  direction

3.  direction

Concept Question: Chrome Inertial Wheel

A fixed torque is applied to the shaft of the chrome inertial wheel. If the four weights on the arms are slid out, the component of the angular acceleration along the shaft direction will

1)  increase.

2)  decrease.

3)  remain the same.

Demo B116: Chrome Inertial Wheel This apparatus is made from four horizontal rods fixed at right angles

attached to an axle and bearing. Four weights can be added at various distances along each rod. A weight is attached to a spring, passed over a pulley and wrapped around the vertical axle of the bearing. The weight exerts a constant torque on the system. The demonstration shows that when the four masses are close to the axle, the angular acceleration is greater than when the four masses are attached to the end of the rods.

Problem Solving Strategy: Two Dimensional Rotation

Step 1: Draw free body force diagrams for each object and indicate the point of application of each force

Step 2: Select point to compute torque about

(generally select center of mass)

Step 3: Apply Newton’s Second Law and Torque Law to obtain equations

Step 4: Look for constraint condition between rotational acceleration and any linear accelerations.

Worked Example: Atwood’s Machine (Torque Method)

A pulley of mass mp, radius R, and moment of inertia Icm about the center of mass, is suspended from a ceiling. An inextensible string of negligible mass is wrapped around the pulley and attached on one end to an object of mass m1 and on the other end to an object of mass m2 , with m1 > m2 . At time t = 0, the objects are released from rest. Find the magnitude of the acceleration of the objects using the torque method.

Table Problem: Moment of Inertia Wheel

A steel washer is mounted on a cylindrical rotor of radius r. A massless string, with an object of mass m attached to the other end, is wrapped around the side of the rotor and passes over a massless pulley. Assume that there is a constant frictional torque about the axis of the rotor. The object is released and falls. As the mass falls, the rotor undergoes an angular acceleration of magnitude α1. After the string detaches from the rotor, the rotor coasts to a stop with an angular acceleration of magnitude α2. Let g denote the gravitational constant.

What is the moment of inertia of the

rotor assembly (including the washer) about the rotation axis?

Demo: Hinged Stick and ball in cup

End of stick accelerates faster than g

24

Concept Question: Conservation Laws

A tetherball of mass m is attached to a post of radius by a string. Initially it is a distance r0 from the center of the post and it is moving tangentially with a speed v0 . The string passes through a hole in the center of the post at the top. The string is gradually shortened by drawing it through the hole. Ignore gravity. Until the ball hits the post,

1.  The energy and angular momentum about the center of the post are constant.

2.  The energy of the ball is constant but the angular momentum about the center of the post changes.

3.  Both the energy and the angular momentum about the center of the post, change.

4.  The energy of the ball changes but the angular momentum about the center of the post is constant.

Concept Question: Conservation laws

A tetherball of mass m is attached to a post of radius R by a string. Initially it is a distance r0 from the center of the post and it is moving tangentially with a speed v0. The string wraps around the outside of the post. Ignore gravity. Until the ball hits the post,

1.  The energy and angular momentum about the center of the post are constant.

2.  The energy of the ball is constant but the angular momentum about the center of the post changes.

3.  Both the energy of the ball and the angular momentum about the center of the post, change.

4.  The energy of the ball changes but the angular momentum about the center of the post is constant.

Demo B114: Skaters Twirl

A person holding dumbbells in his/her arms spins in a rotating stool. When he/she pulls the dumbbells inward, the moment of inertia changes and he/she spins faster.

Concept Question: Twirling Person

A woman, holding dumbbells in her arms, spins on a rotating stool. When she pulls the dumbbells inward, her moment of inertia about the vertical axis passing through her center of mass changes and she spins faster. The magnitude of the angular momentum about that axis is

1.  the same. 2.  larger. 3.  smaller. 4.  not enough information is given to decide.

Concept Question: Figure Skater

A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, she reduces her rotational moment of inertia and her angular speed increases. Assume that her angular momentum is constant. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she has pulled in her arms must be

1.  the same. 2.  larger. 3.  smaller. 4.  not enough information is given to decide.