Chapter 8. Rotational Motion and Equilibrium

8.1. Describing Angular Motion

Describing Angular Motion

• Objects that rotate move in a circular path around a center of rotation.

• To gain a better understanding of rotational motion, we begin by considering the position, speed, and acceleration of a rotating object.

• A coordinate system with an origin at the center of rotation is used to describe the motion of the parts of a rotating object.

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The Polar Coordinate System

Any position can be described in terms of Cartesian coordinates (x,y) or polar coordinates (r, θ).

Polar coordinates are useful for describing rotational motion, with the origin taken as the center of rotation.

Describing Angular Motion As a wheel rotates, every point on the wheel moves in a circular path around the axle, which is the axis of rotation. The angular position of the red dot is the angle θ that it makes with respect to a reference line θ = 0, which indicates how far the dot has rotated.

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The common convention is that positive angles are counterclockwise from the reference line, and negative angles are clockwise.

Arc Length

How can one compute the arc length (=distance) that a rotating particle travels?

Angle Unit Comparision One revolution = 1 rev = 360 degrees = 360⁰

= 2π radians = 2 π rad

1 rad ≈ 57.3 ⁰

The radian is actually

dimensionless, since it is a ratio of

lengths; nevertheless rad is

specified to indicate it is not

deg or rev.

Example

Angular Displacement and Velocity

The angular displacement is the change in angular position (i.e. angle).

Sign of Angular Velocity

Example 8.2

Tangential Speed

Example 8.4

Do children side-by-side on a merry-go-round have the same angular velocity or tangential speed?

Angular Acceleration

Example 8.5

Tangential Acceleration

The tangential acceleration is the

change in tangential speed

per unit time. SI Units: m/s2

Total and Centripetal Acceleration

Even when tangential speed is constant, tangential velocity has changed due to change in direction. This indicates an acceleration.

Summary of Variables

Property Linear Rotational Relation

Position

Velocity

Acceleration

Linear Equation (a = constant)