Introduction: In this lesson, the velocity of a particle traveling on a circular path will be examined. Two different perspectives for measuring a particle's motion are its angular velocity and its linear velocity. During this lesson, we will assume that our objects are moving in uniform circular motion, that is, they are moving along a circular path at a constant speed which is neither increasing nor decreasing.

The Lesson:

The angular velocity of a particle traveling on a circular path is the ratio of the angle traversed to the amount of time it takes to traverse that angle.

1. Consider the Earth which rotates on its axis once every 24 hours. This complete circle is radians. Therefore, the angular velocity of the Earth’s rotation is .



2. A second example is that of a ceiling fan. If the fan rotates 30 times per minute, the angular velocity is radians per minute or .

Angular velocity is a measure of the angular displacement per unit time. Notice that the angular velocity of the fan is much larger than the angular velocity of the Earth. However, the Earth has a much larger radius than a ceiling fan. Therefore a point on the surface of the Earth is moving much faster than the tip of a fan blade. To see this, we will calculate the linear velocity of a point on the surface of the Earth and a point on the tip of a fan blade.

1. The radius of the Earth is approximately 4000 miles. The Earth is rotating at a rate of per hour. Notice that the distance traveled in one hour on the surface of the Earth in 15º of rotation is where we use to calculate the circumference of the Earth and we use as the fraction of the circumference traversed in one hour. The linear velocity is 1047 miles per hour! We only keep from flying off the surface of the Earth because of gravity.



2. To calculate the linear velocity of the fan blade, we note that the blade rotates 30 times in one minute or times per hour. Assuming the radius of the fan is 2 feet, we have the circumference of the fan as . Multiplying this by 1800 gives .

Note: Despite a much larger angular velocity, the tip of the blade of a ceiling fan has a much smaller linear velocity than the Earth because of the vast difference in the lengths of their  respective radii.

We can generalize the calculations made for the Earth and the ceiling fan so that we have formulas which will work for the motion of any particle on a circular path.

1. The angular velocity is a ratio of the total angular measurement through which a particle rotates in a given unit of time. If we use w to stand for angular velocity, we have .

Reviewing the motion of the Earth, recall that the

• Earth has an angular velocity of radians per hour.
• linear velocity of a point on the Earth's surface was calculated by multiplying this angular velocity by the radius of the Earth = .

2. Using this as a guide, we define linear velocity, v, to be where w is angular velocity in radians and r is the radius.

Let's Practice:

1. A Ferris Wheel rotates 3 times each minute. The passengers sit in seats that are 25 feet from the center of the wheel. What is the angular velocity of the wheel in degrees per minute and radians per minute? What is the linear velocity of the passengers in the seats?

3 revolutions per minute is .

The linear velocity is calculated from .

This gives us .

We can change this to miles per hour by multiplying as follows:

.

We can simplify this result by “canceling” the minutes and feet to get

.

2. An object is rotating on a circular path at 4 revolutions per minute. The linear velocity of the object is 400 feet per minute. What is the radius of the circle and what is the angular velocity of the rotating platform?

We first find the angular velocity: 4 revolutions per minute is .

Second, we use . This gives us

.

3. An object moves along a circular path of radius r. What is the effect on the linear velocity if the radius of this circle is doubled?

The linear velocity is . If the linear velocity is recalculated using 2r as the new radius, we have: . Therefore if the radius is doubled, the linear velocity is also doubled.