0.27 Phy1240: circular motion -- speed and velocity  (Page 2/3)

Four sets of horses

Assume that it is a large carousel with horses on four equally spaced concentric circles. You take a seat on one of the horses on the outer circle andyour three friends take seats on horses on the other three circles. The four of you are riding approximately side-by-side.

Constant speed

Once the carousel gets started and comes up to speed, it typically spins at a constant speed for several minutes, after which it slows down andstops.

During the time that the carousel is spinning at constant speed, you and each of your friends would be experiencing uniform circular motion .

Average speed

Let's assume that the radius of the circle on which your horse is positioned is 10 meters. In other words, you are sitting on a horse that is 10 meters fromthe center of the carousel.

The circumference of the circle

Then the circumference of the circle on which you are located would be equal to

c1 = 2*pi*r1 = 2*pi*10m = 62.83 meters

where

• c1 is the circumference of the outer circle
• r1 is the radius of the outer circle
• pi is the mathematical constant pi with a value of 3.14159
• m is distance in meters

Time required to complete one cycle

As the disk spins, you pass the same sighted observer standing close to the carousel again and again. Assume that the sighted observer determines that the time required for you to complete each cyclearound the carousel is 31.41 seconds.

The period

The proper term for the time required for an object to complete one cycle with uniform circular motion is period . Thus, your period would be 31.41 seconds.

Periodic motion

The term period is also used to describe the motion of other objects, such as a pendulum, a rocking chair, etc., resulting in a related term: periodic motion . Periodic motion is motion that is repeated in equal intervals of time. Circular motion is only one of many forms of periodic motion.

Average speed

Getting back to your ride on the carousel, your average speed around the circumference of your circle would beequal to

Avg Speed = distance/time = circumference/time, or

Avg Speed = 2*pi*radius/time, or

Avg Speed = 2*pi*10m/31.41s = 2 m/s

where

• circumference is the circumference of the circle in meters
• Avg speed represents average speed
• time is the number of second required to complete one trip around the circumference; the period

Average speeds of your friends

Now consider the average speed that each of your friends are traveling. Each of you complete one cycle in 31.41 seconds, but your friends don't travel as far as you do inthat amount of time. The circumference of the circles on which they are traveling is smaller than the circumference of the circle on which you aretraveling. Therefore, your average speed is greater than your friends' average speeds.

Assume, for example, that the radii of the circles on which your friends are traveling are 9, 8, and 7 meters respectively. The average speed foryou and each of your friends will be

• s1 = 2*pi*r1/time = 2*pi*10m/31.41s = 2 m/s
• s2 = 2*pi*r2/time = 2*pi*9m/31.41s = 1.8 m/s
• s3 = 2*pi*r3/time = 2*pi*8m/31.41s = 1.6 m/s
• s4 = 2*pi*r4/time = 2*pi*8m/31.41s = 1.4 m/s