Uniform Rotational Motion Study Guide

Introduction

Motion in one dimension are concepts such as displacement, speed and velocity, and acceleration, We will be following these concepts, but in this case, for a circular trajectory, We will end the lesson by studying the centripetal force, acceleration, and the origins of both.

Uniform Circular Motion

First, let us look at the new type of trajectory, a circle, and define some fundamental quantities. We will define uniform circular motion as the motion of an object traveling at a constant speed on a circular path. One rotation around the trajectory is called a revolution.

Uniform Circular Motion

Uniform circular motion is the motion of an object traveling at a constant speed on a circular path.

Angular Displacement

What quantity can describe motion on this circular trajectory? In between the two positions shown in Figure 8.1, there is a distance, arc length, and an angle (revealed by the two radii). The angle described by an object moving around the circular trajectory will be called angular displacement. Also similar to the linear displacement, angular displacement has a positive and negative direction: The positive direction is the counterclockwise rotation, and the negative direction is the clockwise rotation.

Uniform Rotational Motion Figure 8.1

When the angular displacement is small, we can determine the angle in radians as:

where s is the arc length and r is the radius.

The conversion factor between radians and degrees can be found if we think of 2π radians that are equivalent to 360°. Then:

2π radians = 360°

1 radian = 57.3°

Example

A rock at the end of a string is spun around in a circle of radius 0.50 m. Between two close instances, the arc length is 10.0 cm. Find the angular displacement in both radians and degrees.

Solution

First, convert the quantities to SI. Next, list the data and solve the problem.

r = 0.5 m

s = 10 cm = 10 cm · 1 m/100 cm = 0.1 m

θ = ?

Tangential Velocity

Although the uniform circular motion is characterized by constant speed, the direction of motion changes all the time, and therefore, the velocity will have different directions, as is shown in Figures 8.3 and 8.4. At two different times, the object occupies different positions on the circle. This defines the linear displacement, and by dividing it by time, we can determine the linear velocity. In Figure 8.3, we are shown two distant positions on the circle and the corresponding displacement, d, whereas in Figure 8.4, the displacement is very small, d → 0.

Uniform Rotational Motion Figure 8.3

Uniform Rotational Motion Figure 8.4

As the two dotted lines show, the change in position modifies the direction of the displacement, and for a very small interval of time, Δt → 0, displacement becomes tangent to the radius and so does the velocity. So, at every point on the trajectory, the velocity will be tangent to the circle pointing in the direction of motion and defining tangential velocity.

The magnitude of the average velocity (for uniform circular motion, this is also the value of the instantaneous speed) is given by:

And the direction is tangent to the trajectory at every point. For one revolution, the time is called a period, and its symbol is T. T is measured in seconds (s).

The inverse of the period is called the frequency and defines the number of rotations per second. Frequency is measured in s^–1 lor hertz (Hz), named for Heinrich Rudolf Hertz (1857–1894), and its symbol is f.

Then the expression of the speed becomes:

v = 2 · π · r · f

Tangential Velocity