# Rotation

## Proof of the Parallel-Axis Theorem

Proof of the Parallel-Axis Theorem Let 0 be the center of mass of the arbitrarily shaped body shown in cross section in . Place the origin of the coordinates at O. Consider an axis through 0 perpendicular to the plane of the figure. and another axis through point P parallel to the first axis. Let the x and …

## Parallel Axis Theorem

Parallel Axis Theorem Suppose we want to find the rotational inertia I of a body of mass M about a given axis. In principle, we can always find I with the integration. However,  there is a shortcut if we happen to already know the rotational inertia loom of the body about a parallel axis that extends through …

## Calculating the Rotational Inertia

Calculating the Rotational Inertia If a rigid body consists of a few particles. we can calculate its rotational inertia about a given rotation axis with  that is we can find the product for each particle and then sum the products. Gives the results of such integration for nine coinmon body shapes and the indicated axes of rotation.

## Kinetic Energy Of Rotation

Kinetic Energy Of Rotation The rapidly rotating blade of a table saw certainly has kinetic energy due to that rotation. How can we express the energy? We cannot apply the familiar formula  to the saw as a whole because that would only give us the kinetic energy of the saw’s center of mass, which is zero. We noted …

## PROBLEM SOLVING TACTICS

PROBLEM SOLVING TACTICS Tactic 1: Units for Angular Variables We began the use of radian measure for all angular variables whenever we are using equations that contain both angular and linear variables. Thus, we must express angular displacements in radians, angular velocities in rads and rad/min, and angular accelerations ,  are marked to emphasize this. The only exceptions to this …

## Sample Problem

Sample Problem Shows a centrifuge used to accustom astronaut trainees to high accelerations. The radius r of the circle traveled by an astronaut. (a) At what constant angular speed must the centrifuge rotate if the astronaut is to have a linear acceleration of magnitude. SOLUTION: The Key Idea is this.Because the angular speed is constant, the angular acceleration. (b) …

## The Acceleration

The Acceleration Here we run up against a complication. In represents only the part of the linear acceleration that is responsible for changes in the magnitude v of the linear velocity v. Like V. that part of the linear acceleration is tangent to the path of the point in question. Thus.  the linear acceleration of a point on …

## The Speed

The Speed However, is the linear speed (the magnitude of the linear velocity) of the point in question,  is the angular speed ta of the rotating body. Caution: The angular speed w must be expressed in radian measure. Equation 11-18 tells us’ that since all points within the rigid body have the same angular speed w, points …

## The Position

The Position If a reference line on a rigid body rotates through an angle 0, a point within the body at a position r from the rotation axis moves a distance s along a circular. This is the first of our linear-angular relations. Caution: The angle 0 here must be measured in radians because is itself the …

## Relating the Linear and Angular Variables

Relating the Linear and Angular Variables We discussed uniform circular motion, in which a particle travels at constant linear speed v along a circle and around an axis of rotation. When a rigid body, such as a merry go round, rotates around an axis, each particle” in the body moves in its own circle around that axis. Since the …