0.32 Phy1320: angular momentum -- the mathematics of torque  (Page 2/13)

A review of rotational kinematics

One of the objectives of this review is to summarize the angular kinematic variables that are used to describe rotational motion and relate themto the translational kinematic variables that we already know about.

Given a system of particles, we can describe motion as having two components:

• The motion of the center of mass.
• Motion relative to the center of mass.

Terminology

When an object rotates, it experiences an angular displacement, which I will refer to as theta in this review.

(Textbooks typically use the Greek letter theta for this purpose. However, your Braille display probably won't display the Greek letter theta. Therefore, Iwill spell it out when I use it in text, and will replace it by the character "Q" when I use it in an equation.)

The time rate of change of the angular displacement is the angular velocity, which I will refer to as omega in thisreview.

The time rate of change of the angular velocity is the angular acceleration, which I will refer to as alpha in this review.

Similarity to translational motion

These definitions are very similar to definitions from earlier modules having to do with translational motion. For example, when an object moves, itexperiences a displacement. The time rate of change of the displacement is the velocity, and the time rate of change of the velocity is the acceleration.

This similarity derives from the fact that rotational motion can be described by a single angular displacement, theta, just as linear motion can be described by a single spatialdisplacement, x.

Constant angular acceleration

For constant angular acceleration, we can derive a set of equations that are analogous to corresponding equations for translational motion:

Q = Q0 + w0*t + (1/2)*a*t^2

w = w0 + a*t

w^2 = w0^2 + 2*a*(Q - Q0)

where

• Q represents the angular displacement theta
• Q0 represents the initial angular displacement
• w represents the angular velocity omega
• w0 represents the initial angular velocity
• a represents the angular acceleration alpha
• t represents time in seconds

Corresponding translational equations

For reference, here are the three translational equations that correspond to the equations given above . These equations were explained in an earlier module that involved the constant translational acceleration of gravity.

h = h0 v0*t + 0.5*g*t^2

v = v0 + g*t

v^2 = v0^2 + 2*g*(h-h0)

where

• h represents the translational displacement
• h0 represents the initial translational displacement
• v represents the translational velocity
• v0 represents the initial translational velocity
• g represents the acceleration of gravity
• t represents time in seconds

Motion of a point that is a fixed distance from the rotational axis

It is customary to define counter clockwise rotation as the positive direction of rotation. That will be the case in this review.

Consider a rotating disk. Since all points on the disk are rotating together, we candetermine the linear displacement, speed and acceleration of any point on the disk in termsof the corresponding angular parameters.