0.29 Phy1260: circular motion -- the mathematics of circular motion  (Page 6/7)

Example scenarios

In this section, I will work through some examples that illustrate what you learned in the earlier section along with what you have learned in earlier modules.

Circumference of the Earth at the equator

The radius of the Earth at the equator is equal to approximately 6378km. What is the circumference of the earth at the equator.

Solution:

If you were to travel around the Earth at the equator, you would travel along a circular arc that subtends an angle of 2*pi radians. We know how to computethe length of the circular arc given the radius and the subtended angle:

arc length = (subtended angle) * radius, or

circumference = 2*pi*6378km = 40074 km

Speed of a point on the equator

The Earth rotates around its axis once each 24 hours. Therefore, a point on the equator makes one full trip around a circle with the circumference of theEarth each 24 hours.

Assume you are standing at a point on the equator. Ignoring all of the other motions of the universe, what is the speed with which you are traveling aroundthat circle?

What is the angular velocity of the earth in radians per second.

Does the angular velocity of the Earth change when you drive North from the equator?

Solution A

Speed

Since we already know the circumference of the Earth, we know that you will travel 40074 km each 24 hours. Therefore,

speed = 40074km/24hr = 463.8 meters/second, or

speed = 1037 miles/hour

Did you know that you are constantly moving through space at a speed slightly greater than 1000 miles per hour?

Angular velocity

We also know that the earth rotates around its axis by 2*pi radians each 24 hours. Therefore, the angular velocity of the earth is

w = 2*pi radians/24 hours = 7.27 *10^(-5) radians / second

Differences in angular velocity

For purposes of this discussion, the Earth does not distort as it rotates. Therefore, the angular velocity of every point on the surface of the Earthrotates around the Earth's axis with the same angular velocity; namely 2*pi radians every 24 hours.

However I did see on TV the other day that the angular velocity of the Earth's core may be different from the angular velocity of the outer crust ofthe Earth. Apparently this can happen because the core is not firmly attached to the crust, but rather is suspended in a bath of molten rock.

Solution B

You learned earlier that the speed is equal to the product of the angular velocity and the radius. Therefore,

speed = w * r, or

speed = (7.27 *10^(-5) radians / second) * 6378km, or

speed = 463.7 m/s

which agrees with Solution A above.

Note that w represents angular velocity and r represents radius in the above calculations.

Period and frequency

A child's toy contains a round disk that rotates with a period of 0.628 seconds.

What is the frequency with which a spot on the edge of the disk passes a fixed mark on the body of the toy.

What is the angular velocity of the toy?

Solution:

f = 1/T = 1/(0.628) = 1.59 Hz

w = 2*pi*f = 10 radian/second

where

• f represents frequency
• T represents the period
• w represents angular velocity