Enduring Understanding 4.D
A net torque exerted on a system by other objects or systems will change the angular momentum of the system.
Essential Knowledge 4.D.1
Torque, angular velocity, angular acceleration, and angular momentum are vectors and can be characterized as positive or negative depending on whether they give rise to or correspond to counterclockwise or clockwise rotation with respect to an axis.
Essential Knowledge 4.D.2
The angular momentum of a system may change due to interactions with other objects or systems.
Essential Knowledge 4.D.3
The change in angular momentum is given by the product of the average torque and the time interval during which the torque is exerted.
Angular momentum is the rotational equivalent to linear momentum. It is represented by the equation L=I𝜔 where L is the angular momentum, I is the moment of inertia of the object, and 𝜔 is the angular velocity of the object. In addition, the angular momentum of an object moving in a straight line relative to a fixed point can be found by multiplying its linear momentum (p) by the perpendicular distance (r) to the point as shown in the picture to the right.
Image courtesy of MilesMathis.
Just like a force acting on an object causes a change in linear momentum, a net torque causes a change in the angular momentum of the object.
EXAMPLE: (AP Classroom)
The left end of a rod of length d and rotational inertia I is attached to a frictionless horizontal surface by a frictionless pivot, as shown above. Point C marks the center (midpoint) of the rod. The rod is initially motionless but is free to rotate around the pivot.
A student will slide a disk of mass:
toward the rod with velocity v0 perpendicular to the rod, and the disk will stick to the rod a distance x from the pivot.
The student wants the rod-disk system to end up with as much angular speed as possible.
Briefly explain your reasoning without manipulating equations.
CORRECT ANSWER: To the right of C
REASONING: To make the largest angular speed, the rod needs to get the greatest angular momentum possible (L=I𝜔). To get the largest angular momentum, the rod must be given the greatest possible torque. Since torque depends on the force and the radius from the pivot, hitting the rod at the greatest radius will produce the greatest torque. Because the pivot is on the left side of C, we want to hit on the right side of C.
ALTERNATE EXPLANATION: To make the largest angular speed, the rod needs to get the greatest angular momentum possible (L=I𝜔). To do this, we want the disk to transfer the greatest amount of momentum to the rod. Since angular momentum is conserved in the rod-disk system (no outside torques), this means that we want the initial angular momentum of the disk to be a maximum. The angular momentum of the disk is equal to L=mvr where r is the distance from the pivot.
the largest it can possibly be, it needs to hit at the greatest distance away from the pivot which is on the right side of C.
(This explanation uses the law of conservation of momentum which will be covered in the next section.)
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