\Rotational Motion

Rotational Inertia and Newton’s Second Law  In linear motion, net force and mass determine the acceleration of an object.  For rotational motion, torque determines the rotational acceleration.  The rotational counterpart to mass is rotational inertia or moment of inertia. Just as mass represents the resistance to a change in linear motion, rotational inertia is the resistance of an object to change in its rotational motion. Rotational inertia is related to the mass of the object. It also depends on how the mass is distributed about the axis of rotation.

Rotational Inertia and Newton’s Second Law  The resistance to a change in rotational motion depends on: the mass of the object; the square of the distance of the mass from the axis of rotation.  For an object with its mass concentrated at a point: Rotational inertia = mass x square of distance from axis I = mr 2  The total rotational inertia of an object like a merry-go- round can be found by adding the contributions of all the different parts of the object.

Rotational Inertia and Newton’s Second Law  Newton’s second law for linear motion: F net = ma  Newton’s second law for rotational motion: The net torque acting on an object about a given axis is equal to the rotational inertia of the object about that axis times the rotational acceleration of the object.  net = I  The rotational acceleration produced is equal to the torque divided by the rotational inertia.

Example: a baton with a mass at both ends  Most of the rotational inertia comes from the masses at the ends.  A torque can be applied at the center of the rod, producing a rotational acceleration and starting the baton to rotate.  If the masses were moved toward the center, the rotational inertia would decrease and the baton would be easier to rotate.

Conservation of Angular Momentum How do spinning skaters or divers change their rotational velocities? I = mr 2

Angular Momentum  Linear momentum is mass (inertia) times linear velocity:p = mv  Angular momentum is rotational inertia times rotational velocity: L = I  Angular momentum may also be called rotational momentum. A bowling ball spinning slowly might have the same angular momentum as a baseball spinning much more rapidly, because of the larger rotational inertia I of the bowling ball.

Conservation of Angular Momentum  Linear momentum is conserved if the net external force acting on the system is zero.  Angular momentum is conserved if the net external torque acting on the system is zero.

Angular momentum is conserved by changing the angular velocity  When the masses are brought in closer to the student’s body, his rotational velocity increases to compensate for the decrease in rotational inertia.  He spins faster when the masses are held close to his body, and he spins more slowly when his arms are outstretched.

Angular momentum is conserved by changing the angular velocity  The diver increases her rotational velocity by pulling into a tuck position, thus reducing her rotational inertia about her center of gravity.

Kepler’s Second Law  Kepler’s second law says that the radius line from the sun to the planet sweeps out equal areas in equal times.  The planet moves faster in its elliptical orbit when it is nearer to the sun than when it is farther from the sun.

Kepler’s Second Law  This is due to conservation of angular momentum.  The gravitational force acting on the planet produces no torque about an axis through the sun because the lever arm is zero: the force’s line of action passes through the sun.

Kepler’s Second Law  When the planet moves nearer to the sun, its rotational inertia about the sun decreases.  To conserve angular momentum, the rotational velocity of the planet about the sun must increase.

Angular momentum is a vector  The direction of the rotational-velocity vector is given by the right-hand rule.  The direction of the angular-momentum vector is the same as the rotational velocity.

A student holds a spinning bicycle wheel while sitting on a stool that is free to rotate. What happens if the wheel is turned upside down?  To conserve angular momentum, the original direction of the angular- momentum vector must be maintained.

A student holds a spinning bicycle wheel while sitting on a stool that is free to rotate. What happens if the wheel is turned upside down?  The angular momentum of the student and stool, + L s, adds to that of the (flipped) wheel, - L w, to yield the direction and magnitude of the original angular momentum + L w.

A student sits on a stool holding a bicycle wheel with a rotational velocity of 5 rev/s about a vertical axis. The rotational inertia of the wheel is 2 kg·m 2 about its center and the rotational inertia of the student and wheel and platform about the rotational axis of the platform is 6 kg·m 2. What is the initial angular momentum of the system? a) 10 kg·m 2 /s upward b) 25 kg·m 2 /s downward c) 25 kg·m 2 /s upward d) 50 kg·m 2 /s downward L = I  = (2 kg·m 2 )(5 rev/s) = 10 kg·m 2 /s upward from plane of wheel