The formula
- From linear speed
F_c = mv² / r- From angular speed
F_c = mrω²- Centripetal acceleration
a_c = v² / r- Speed from the period of one revolution
v = 2πr / T
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
F_c | Net inward force required for the circular path | N |
m | Mass of the object | kg |
v | Speed along the circular path | m/s |
r | Radius of the path, measured to the center | m |
ω | Angular speed | rad/s |
a_c | Centripetal acceleration, directed toward the center | m/s² |
T | Period, the time for one full revolution | s |
π | Ratio of a circle's circumference to its diameter | 3.1416 (dimensionless) |
When it applies
- Uniform circular motion, or the instantaneous inward direction at any point on a curved path.
- Work out which real force supplies it before you use the formula. Adding F_c to a free-body diagram alongside that force counts it twice.
- r is the radius of the circular path, measured from the center, not the diameter.
Worked example
Problem. A 0.20 kg puck on the end of a 0.75 m string travels in a circle on a frictionless horizontal table at 3.0 m/s. How hard does the string pull?
- The string tension is the only horizontal force, so it supplies the entire inward force: T = F_c.
- Substitute into F_c = mv² / r: F_c = (0.20 kg)(3.0 m/s)² / 0.75 m.
- (3.0)² = 9.0, so the numerator is (0.20)(9.0) = 1.8.
- Divide: F_c = 1.8 / 0.75 = 2.4 N.
Answer. The string pulls inward with 2.4 N. Doubling the speed to 6.0 m/s would need 9.6 N, four times as much.
Common mistakes
- Drawing an outward centrifugal force on a free-body diagram. In an inertial frame no such force acts on the object.
- Adding F_c as an extra force alongside the tension or friction that is actually producing it.
- Using the diameter of the circle in place of the radius.
- Underestimating how fast the requirement grows. Speed is squared, so a small increase in speed needs a large increase in force.
Related formulas
- Moment of inertia formula:
I = Σ mᵢrᵢ² - Tension formula:
T = mg - Newton's second law:
ΣF = ma