Derivation for Circular Motion Formulas

Deriving a formula for velocity, acceleration, and the centripetal force when in a circular motion...


LEGEND
v = velocity (or speed, if direction is not indicated)
d = displacement (or distance, if direction is not indicated)
t = time (interval)
R = radius
a = acceleration
a_c = centripetal acceleration
F = force
F_net = net force
F_c = centripetal force
m = mass


1.
The distance an object moves in a circular motion is the circumference of the circular motion, which is equal to 2*pi*R. Using the definition of circumference (c=π*2*r) and the definition of velocity (v=d/t), we can derive this formula: v=2*π*R/t.

2.
Draw the circular motion, two radii, and two velocity vectors. Add the two radii vectors to get the net radius. Add the two velocity vectors to get the net velocity.

3.
Since the change in radii over the radius equals the change in velocities over a velocity (ΔR/R = Δv/v), using the definition of velocity (v=ΔR/Δt) and acceleration (a=Δv/Δt), we can derive this formula: a=v^2/R.

4.

Using the acceleration formula that we recently derived (a=v^2/R), we can use substitute velocity with v=2*π*R/t to get a more fancy-looking formula for centripetal acceleration (a=4*π^2*R/T).

5.
Using Newton's Second Law of Motion (F_net=m*a), we can substitute acceleration with a=v^2/R to get the formula for centripetal force (F_c=m*v^2/R), which can be derived even further using v=2*π*R/t to get F_c=4*π^2*R/t^2.

• Centripetal Force lab is due and was handed in today.
• Centripetal Acceleration and Centripetal Force assignments is due tomorrow.
• Next scribe is ERIC.