Remember our description of rotational motion. A rotating object has angular velocity w. Angular velocity has magnitude and direction. The magnitude of the angular velocity is the rotation rate, Dq/Dt. The direction is given by the right-hand rule. The angular acceleration a is defined as the rate of change of the angular velocity. The angular velocity changes when the rotation rate is increasing or decreasing and when the axis of rotation changes direction.
Assume an object has angular velocity w about an axis. We define the angular momentum L of the object about this axis as L = Iw. It is a measure of an object's rotational motion about the axis. The angular momentum L is the product of the object's moment of inertia I times its angular velocity w about the chosen axis. Angular momentum has magnitude and direction. The direction of the angular momentum of an object rotating about one of its symmetry axes is the direction of the angular velocity (given by the right hand rule).
To produces an angular acceleration a about an axis, a torque t = I * a is required.
torque = moment of inertia times angular acceleration
Using a = Dw/Dt
we can rewrite the expression for the torque as t = I *a = I*Dw/Dt = DL/Dt.torque = change of angular momentum / time it takes to make this change
The equation t = DL/Dt. is the rotational analog of Newton's second law, F = Dp/Dt. If a torque t acts for a time interval Dt, the angular momentum changes by
D
L = tDtIf an object has many independently rotating parts, the total angular momentum of the object is the sum of the angular momenta of all its parts.
The total angular momentum of a a single object is constant if no external torque acts on the object. An object cannot exert a torque on itself. The total angular momentum of two interacting objects is also constant if no external torque acts on the objects. Newton's third law tells us the forces the objects exert on each other are equal in magnitude and opposite in direction. The interaction forces produce torques equal in magnitude and opposite in direction. These torques change the angular momentum of each object by the same amount, but the changes will have opposite directions. When we sum them up to find the change in the total angular momentum, we obtain zero.
If no external torque acts on a system of interacting objects, then their total angular momentum is constant
In the video clip shown below the total angular momentum of the system points upward. The person is stopping a spinning wheel and the stool starts to spin.
 Some of the first few frames: |
As the person applies a torque to the wheel, the wheel applies a torque to the person. The magnitudes of the angular momenta of the wheel and of the person change at the same rate, but their sum remains constant.
Now let us look at the football thrown by the right-handed quarterback. It spins to the right. Initially its angular momentum and velocity vector point in the same direction.
 |
Were it not for the air flow around the ball, the spin axis, i.e. the direction of the angular momentum L of the ball, would be fixed in space. Along the ball’s trajectory the directions of the angular momentum L and the velocityr v would not be parallel, since the direction of the velocity is changing. |
| But you have probably
observed, that when a pass is perfectly thrown and the football spins about its
symmetry axis, the football tilts during flight, so that its symmetry axis is
tangent to its trajectory.
Why? |