Centripetal force appears whenever an object moves along a circular path. Even if the speed is constant, the velocity is
not constant because its direction changes continuously. This changing direction requires an acceleration toward the center
of the circle. That inward acceleration is called centripetal acceleration.
Velocity, angular speed, and period
Uniform circular motion can be described using linear speed \(v\), angular speed \(\omega\), or period \(T\). Linear speed
tells how fast the object moves along the circular path. Angular speed tells how quickly the angle changes. Period tells
how long one complete revolution takes.
Linear and angular speed.
\[
\begin{aligned}
v &= \omega r
\end{aligned}
\]
The angular speed and period are related by:
Angular speed from period.
\[
\begin{aligned}
\omega &= \frac{2\pi}{T}
\end{aligned}
\]
Combining these gives the speed from the period:
Linear speed from period.
\[
\begin{aligned}
v &= \frac{2\pi r}{T}
\end{aligned}
\]
Centripetal acceleration
The acceleration in uniform circular motion points inward, not forward along the path. Its magnitude can be written in
several equivalent ways:
Centripetal acceleration formulas.
\[
\begin{aligned}
a_c &= \frac{v^2}{r} \\
&= \omega^2 r \\
&= \frac{4\pi^2 r}{T^2}
\end{aligned}
\]
These equations show two important patterns. First, increasing speed strongly increases centripetal acceleration because
\(a_c\) depends on \(v^2\). Doubling the speed makes \(a_c\) four times larger. Second, for the same speed, a larger radius
reduces centripetal acceleration because the path curves more gently.
Centripetal force
Centripetal force is not a separate new force. It is the name given to the inward net force required to produce circular
motion. Depending on the situation, the real inward force may be tension, friction, gravity, a normal force component, or
a combination of several forces.
Centripetal force.
\[
\begin{aligned}
F_c &= ma_c \\
&= m\frac{v^2}{r} \\
&= m\omega^2 r \\
&= m\frac{4\pi^2r}{T^2}
\end{aligned}
\]
The direction of \(F_c\) is always toward the center of the circular path. The velocity vector is tangent to the circle and
perpendicular to the inward acceleration vector. This is why an object can move at constant speed while still accelerating:
the acceleration changes the direction of velocity, not its magnitude.
Common examples
A ball on a string is held in circular motion by string tension. A car turning on a flat road is held by static friction
between the tires and road. A satellite orbiting Earth is held by gravity. In all three cases, the inward net force must
match \(m v^2/r\). If the available inward force is too small, the object cannot follow the intended circular path.
| Known quantity |
Useful relation |
Then compute |
| Linear speed \(v\) |
\(a_c=v^2/r\) |
\(F_c=ma_c\) |
| Angular speed \(\omega\) |
\(v=\omega r,\quad a_c=\omega^2r\) |
\(F_c=m\omega^2r\) |
| Period \(T\) |
\(\omega=2\pi/T,\quad v=2\pi r/T\) |
\(F_c=4\pi^2mr/T^2\) |
| Frequency \(f\) |
\(\omega=2\pi f\) |
\(a_c=4\pi^2f^2r\) |
This calculator assumes uniform circular motion, so the speed magnitude is constant. If the speed changes, tangential
acceleration must be added separately from the inward centripetal acceleration.
