In the final steps we express the above result concisely, replacing the sum in the last equation by the single quantity I.
Step Reason
6. I = Ȉmiri
2
moment of inertia
7. L = IȦ substitute equation 6 into equation 5
10.10 - Torque and angular momentum
A net force changes an object’s velocity, which means its linear momentum changes as
well. Similarly, a net torque changes a rotating object’s angular velocity, and this
changes its angular momentum.As Equation 1 shows, the product of torque and an interval of time equals the change in
angular momentum. This equation is analogous to the equation from linear dynamics
stating that the impulse (the product of average force and elapsed time) equals the
change in linear momentum.
In the illustration to the right, we show a satellite rotating at a constant angular velocity
and then firing its thruster rockets, which changes its angular momentum. The thrusters
apply a constant torque IJ to the satellite for an elapsed time ǻt. In Example 1, the
satellite’s change in angular momentum is calculated using the equation. (The change
in the satellite’s mass and moment of inertia resulting from the expelled fuel are small
enough to be ignored.)Torque and angular momentum
IJǻt = ǻL
IJ = torque
ǻt = time interval
ǻL = change in angular momentum
The rockets provide 56 N·m of
torque for 3.0 s. What is the
amount of change in the
satellite's angular momentum?
IJǻt = ǻL
(56 N·m)(3.0 s) = ǻL
ǻL = 168 kg·m^2 /s
10.11 - Conservation of angular momentum
Linear momentum is conserved when there is no external net force acting on a system. Similarly, angular momentum is conserved when there
is no net external torque. To put it another way, if there is no net external torque, the initial angular momentum equals the final angular
momentum. This is stated in Equation 1.
The principle of conservation of linear momentum is often applied to collisions, and the masses of the colliding objects are assumed to remain
constant. However, with angular momentum, we often examine what occurs when the moment of inertia of a body changes. Since angular
momentum equals the product of the moment of inertia and angular velocity, if one of these properties changes, the other must as well for the
angular momentum to stay the same. This principle is used both in classroom demonstrations and in the world of sports. In a common
classroom demonstration, a student is set rotating on a stool. The student holds weights in each hand, and as she pushes them away from her
body, she slows down. In doing so, she demonstrates the conservation of angular momentum: As her moment of inertia increases, her angularAngular momentum conserved
No external torque
Angular momentum is constant(^198) Copyright 2007 Kinetic Books Co. Chapter 10