Angular Quantities

Torque (m N), Moment of Inertia (kg m^2), Angular Acceleration (rad/s^2) replace Force, Mass, Acceleration

• Torque is product of moment-arm r and force perpendicular to moment-arm:  `tau = r * F
• Torque has a direction given by the right-hand rule.
• Moment of inertia is sum of mr^2 (see below)

If we know the # of revolutions and final velocity is zero, to find angular accel `alpha:

• Analyze velocities:
• Find ave angular vel in rev / sec ( `omegaAve = angular displacement / time required)
• Since 1 rev = 2 `pi rad, multiply by 2 `pi rad / rev to get `omegaAve in rad / s.

• Find accel `alpha = change in ang vel / time required [ `alpha = `d`omega / `dt ]
• Since `omegaFinal = 0, if angular accel is uniform we see that `omegaInitial = 2 * `omegaAve.
• Thus `alpha = ( `omegaFinal - `omegaInitial) / (`dt).

If we now know the moment of inertia we can find the torque required to produce this angular acceleration:

• Newton's 2d Law:  `tau = I `alpha

We find moment of inertia by adding up mr^2 for every particle of the object:

• m is particle mass, r is distance from axis of rotation
• if the object is a uniform disk of radius R with mass M we have I = .5 M R^2
• if the object is a uniform sphere of radius R with mass M we have I = .4 M R^2
• if the object is a uniform thin rod of length L with mass M, pivoted about its center, we have I = 1/12 M L^2
• if the object is a uniform thin rod of length L with mass M, pivoted about an end, we have I = 1/3 M L^2 (note 4-1 ratio with previous since everything is twice as far, on average, from axis of rotation and since doubling L fourples L^2)

• If a system is made up of a number of objects and particles, all rotating about the same axis, the individual moments of inertia add up to the total moment of inertia.

Angular Kinetic Energy is .5 I `omega^2

• for a single particle .5 I `omega^2 = .5 m v^2, so when we add up all the m v^2 we add up all the I `omega^2 giving the total I `omega^2.

Angular Momentum is I `omega, and is conserved in isolated systems

• If I changes and no external force is exerted on the system, I `omega can't change.   So if I increases `omega decreases, and vice versa.
• The direction of angular velocity is given by the right-hand rule.