| 2πr |
 |
| T |
Uniform circular motion: motion in a circular path at constant speed.
r = the radius of the circular path
T = the period, the time to go around once
| v | = |
|
As in straight-line motion, the relationship between a and v is the same as that between v and r:
| a | = |
|
Combining these two equations gives us:
| centripetal acceleration: | ac | = |
|
For motion on circular paths it can be useful to describe motion using angular variables. Instead of asking how much distance has been covered, we sometimes ask how much of an angle something has spun through. There are equivalent questions for velocity and acceleration.
Distance: s = rθ
Velocity: v = rω
Acceleration: at = rα
This acceleration involves a speeding up or slowing down of an object as it moves along a circular path, and is equal to zero for uniform circular motion. The a is in a direction tangent to the circle, so its the tangential acceleration. This is very different from the centripetal acceleration, which acts in the radial direction.
Consider the Earth orbiting the Sun. The only interaction we've got to worry about is the force of gravity. The Sun exerts a gravitational force on the Earth that points toward the Sun.
Knowing the distance to the Sun and how long it takes the Earth to orbit the Sun, we can calculate the acceleration the Earth is experiencing.
r = 150 million km = 1.5 x 1011 m
T = 1 year = π x 107 s
| v | = |
|
= |
|
= | 30000 m/s | = | 30 km/s |
| ac | = |
|
= |
|
= |
|
= | 6 x 10-3 m/s2 |
Pretty small, but just right to keep us in orbit.
The centripetal acceleration is the special form the acceleration has when an object is experiencing uniform circular motion. It is:
| ac | = |
|
and is directed toward the center of the circle.
Newton's second law can then be written as:
| ΣF | = | ma | = |
|