To find a particular song on your MP3/MP4 Player, you may use a scroll wheel. This involves moving your finger around the wheel in a circular motion. Unfortunately for you, the song you want is near the very bottom of your songs list. Since these media players can often hold over 1,000 songs, you have to scroll fast! As you are moving your finger in a circle, you might wonder if you could measure how fast your finger is covering the distance around the circle.
Watching your finger, you realize that your finger is moving around the circle twice every second. If the radius of the scroll wheel is 2 cm, what is the angular velocity of your finger as you scroll through your songs list? What is the linear velocity?
You may already be familiar with the measurement of speed as the relationship of an object's distance traveled to the time it has been in motion. However, this relationship is for objects that are moving in a straight line. What about objects that are traveling on a circular path?
Do you remember playing on a merry-go-round when you were younger?
If two people are riding on the outer edge, their velocities should be the same. But, what if one person is close to the center and the other person is on the edge? They are on the same object, but their speed is actually not the same.
Look at the following drawing.
Imagine the point on the larger circle is the person on the edge of the merry-go-round and the point on the smaller circle is the person towards the middle. If the merry-go-round spins exactly once, then both individuals will also make one complete revolution in the same amount of time.
The formula for angular velocity is:
In order to know the linear speed of the particle, we would have to know the actual distance, that is, the length of the radius. Let’s say that the radius is 5 cm.
Remember in a unit circle, the radius is 1 unit, so in this case the linear velocity is the same as the angular velocity.
Here, the distance traveled around the circle is the same for a given unit of time as the angle of rotation, measured in radians.
Calculating the Linear and Angular Velocity
1. Lindsay and Megan are riding on a Merry-go-round. Megan is standing 2.5 feet from the center and Lindsay is riding on the outside edge 7 feet from the center. It takes them 6 seconds to complete a rotation. Calculate the linear and angular velocity of each girl.
As we discussed previously, their linear velocities are different. Using the formula, Megan’s linear velocity is:
Lindsay’s linear velocity is:
We know that the equation for angular velocity is
We can use the given equation to find his linear velocity:
Solving for Unknown Values
How long does it take the bug in the previous problem to go through two complete turns?
Earlier, you were asked to find the angular and linear velocity of your finger.
As you found out in this section, the angular velocity is the change in angle divided by the change in time. Since you sweep around the circle twice in a second, this becomes:
Further, you can find the linear velocity with the equation:
Doris and Lois go for a ride on a carousel. Doris rides on one of the outside horses and Lois rides on one of the smaller horses near the center. Lois’ horse is 3 m from the center of the carousel, and Doris’ horse is 7 m farther away from the center than Lois’. When the carousel starts, it takes them 12 seconds to complete a rotation.
Calculate the linear velocity of each girl. Calculate the angular velocity of the horses on the carousel.
How long does it take a proton to make a complete rotation around the collider? What is the approximate (to the nearest meter per second) angular speed of a proton traveling around the collider? Approximately how many times would a proton travel around the collider in one full second?
Ted is standing 2 meters from the center of a merry go round. If his linear velocity is 6 m/s, what is his angular velocity?
Beth and Steve are on a carousel. Beth is 7 ft from the center and Steve is right on the edge, 7 ft further from the center than Beth. Use this information and the following picture to answer questions 1-6.
- The carousel makes a complete revolution in 12 seconds. How far did Beth go in one revolution? How far did Steve go in one revolution?
- If the carousel continues making revolutions every 12 seconds, what is the angular velocity of the carousel?
- What are Beth and Steve's linear velocities?
- How far away from the center would Beth have to be in order to have a linear velocity of
πft per second.
- The carousel changes to a new angular velocity of
π3radians per second. How long does it take to make a complete revolution now?
- With the carousel's new velocity, what are Beth and Steve's new linear velocities?
- Beth and Steve go on another carousel that has an angular velocity of
π8radians per second. Beth's linear velocity is 2πfeet per second. How far is she standing from the center of the carousel?
- Steve's linear velocity is only
π3feet per second. How far is he standing from the center of the carousel?
- What is the angular velocity of the minute hand on a clock? (in radians per minute)
- What is the angular velocity of the hour hand on a clock? (in radians per minute)
- A certain clock has a radius of 1 ft. What is the linear velocity of the tip of the minute hand?
- On the same clock, what is the linear velocity of the tip of the hour hand?
- The tip of the minute hand on another clock has a linear velocity of 2 inches per minute. What is the radius of the clock?
- What is the angular velocity of the second hand on a clock? (in radians per minute)
- The tip of the second hand on a clock has a linear velocity of 2 feet per minute. What is the radius of the clock?
To see the Review answers, open this PDF file and look for section 2.8.