The Children’s Museum in Indianapolis, Indiana, houses an antique carousel.  The carousel contains three concentric circles of 

The inner circle of animals is approximately 11 ft. from the center, and the outer circle of animals is approximately 20 ft. from the center.  The carousel makes 2⁵/₈ rotations per minute.  In a later examination, using new information, you can determine the angular and linear velocities of someone riding an animal in the inner circle and of someone riding an animal in the same row in the outer circle.  

This carousel is a circular object that turns around an axis, found in its center.  Ferris wheels, gears, tires and compact discs (remember those?) are other examples of this setup.  As any of these objects rotate counter-clockwise about the center, an object at the edge moves through an angle relative to its starting position known as 

, another way of stating the angle of rotation.

 at the origin of a coordinate plane, and a point 

 on the circle rotating counter-clockwise.  Let the positive x-axis then, having point 

 on the circle, serve as the initial side.  The terminal side is known as ray 

and the angle produced is labeled θ.  The measure of θ changes as 

 moves around the circle creating that angular displacement of θ, and all points on ray 

 move through the same angle per unit of time.

Quite simply, then, the ratio of the change in the central angle to the time required for the change is known as 

, usually represented in Mathematics by the lowercase Greek letter ω (omega).  So ,

, where θ is the displacement measured in radians.

a) What would the angular displacement in radians be for 4.5 revolutions of an object?  Round to the nearest tenth.

b) What would the angular velocity be for an object to complete 7.3 revolutions in 5 seconds?  Round to the nearest tenth.

Let’s look back at the carousel and begin looking at the angular velocity for each rider in radians per second by using a procedure called 

.  In this process, unit labels are treated as mathematical variables and can be divided out.  For instance, if the carousel makes 2⁵/₈ (2.625) revolutions per minute (rpm), we can immediately convert rpm to radians per second (rps):

       1 minute                    60 seconds          1 revolution

So, if you cross “factor” the revolution units and the minute units, you would end up with the simple equation of (2.625)(1)(2π radians) / (1)(60 seconds)(1) ≊ 0.275 rps.

🔑Determine each angular velocity in RPS. Round to the nearest tenth.

Consider, again, that carousel.  All riders have the same angular velocity.  However, the rider in the outer circle must travel a greater distance than the one in the inner circle.  

 formula from a few lessons ago can come in handy here.  If the object moves with 

) that it travels is given by that formula: 

As that object moves along the circular path, the radius (

) forms a central angle of measure θ.  Since the length of the arc is 

θ (that familiar formula!), the following steps are true to find linear velocity:

The radius remains isolated in the final formula, as it is a constant.  Also, remember, (θ / 

 can be substituted also by our good friend 

g) Using this carousel more thoroughly, use the given information to find out which rider will experience more linear velocity - the one on the outside horse or the one on the inner horse?  Compare your answers with another student.

🔑Find the linear velocity of an object at the given angular velocity and distance from center.  Round to the nearest tenth.

h)  ω = 16.6 radians per second, r = 8 cm

i)  ω = 6.1π radians per minute, r = 1.8 m

Solve the following prompts using any of the information from this card or previous lessons.  Some prompts will indicate hints.  Round to the nearest thousandth when necessary.

 of a Ferris wheel is 80 ft. If the Ferris wheel makes one revolution every 45 seconds, find the linear velocity of a person riding in a gondola.

m) Suppose that the linear velocity of a person riding on that Ferris wheel is 8 ft. per second.  What is the time for one revolution of the Ferris wheel?

n) The tires on a race car have a diameter of 30 in.  If the tires are turning at a rate of 2000 revolutions per minute, determine the race car’s speed in miles per hour (mph).  HINT: Use the process of 

 to find the right units and conversions.

o) Space, the final frontier ™.  The radius of Mars is roughly 3375 km. and its time of one rotation on its axis is 24.623 in hours (

).  Earth, one planet closer to the sun, has a radius of 6356 km. and completes one rotation in 23.935 h.  Find and compare the linear velocity of a point on the equator of Mars to a point on the equator of Earth.

p)  Low Earth orbiting (LEO) satellites orbit between 200 and 500 miles above Earth.  in order to keep the satellites at a constant distance from Earth, they must maintain a speed of 17,000 mph.  Assume that Earth’s radius for this example is 3960 miles.  What is the angular velocity needed to maintain a LEO satellite at 200 miles above Earth?

p.5) How far above Earth is a LEO with an angular velocity of 4 radians per hour?

q) 🦄Two children are playing on the seesaw.  The lighter child is 9 ft. from the fulcrum and the heavier is 6 ft. from the fulcrum.  As the lighter child goes from the ground to the highest point, she travels through an angle of 35° in ½ second.  What is the angular velocity and the linear velocity of the lighter and the heavier child, respectively?

Students will apply knowledge of Trigonometry and the unit circle to compare angular and linear velocity.

He's Moving From the Outside...