Free Centripetal Force Calculator

Centripetal Force in Circular Motion

The Centripetal Force Calculator is a free online physics tool that streamlines calculations for any object moving along a curved path. Serving as both a centripetal force calculator and a centripetal acceleration calculator, this radial force calculator allows you to quickly determine the required centripetal force, acceleration, or velocity by entering a few values. It supports multiple unit systems and automatically converts between them, making it a versatile circular motion calculator.

Centripetal Force Definition and Formula

Centripetal force is the inward force that must act on a body to keep it moving in a circle. It always points toward the center of the curve. The standard centripetal force formula is:

$$F = \dfrac{m v^{2}}{r}$$

where $F$ is the centripetal force, $m$ is the mass, $v$ is the linear velocity, and $r$ is the radius of the circular path.

Using the relationship between linear and angular velocity $v = \omega r$, the equation can be expressed in terms of angular velocity $\omega$:

$$F = m \omega^{2} r$$

The centripetal acceleration, derived from Newton's second law, is $a = v^{2} / r$.

Units of Centripetal Force

Since centripetal force is a type of force, it uses the same units as any other force:

• SI: newton (N)
• Imperial (foot‑pound‑second): poundal (pdl)
• English Engineering: pound‑force (lbf)
• CGS: dyne (dy)

The calculator handles all these units, so you can input values in your accustomed system and instantly read the result in another. Centripetal acceleration is expressed in metres per second squared (m/s²).

Centripetal vs. Centrifugal Force

A frequent point of confusion is the difference between centripetal and centrifugal force. While they share the same magnitude $m v^{2} / r$, they arise from different frames of reference. In an inertial frame (e.g., a stationary observer watching a child on a merry‑go‑round), only the centripetal force acts toward the center. In a rotating non‑inertial frame (the child's perspective), an apparent centrifugal force seems to push outward, balancing the centripetal force. The centripetal force is real in both frames, whereas the centrifugal force is a fictitious or inertia force that exists solely in the rotating frame.

Worked Examples

Example 1 – Force on a car rounding a curve
A car of mass $2000\ \text{kg}$ goes around a circular track of radius $10\ \text{m}$ at a speed of $12.5\ \text{m/s}$. Applying the centripetal force equation:

$$F = \dfrac{2000 \times (12.5)^{2}}{10} = 31\,250\ \text{N} \;\;(31.25\ \text{kN})$$

Example 2 – Determining velocity from known force
An object with a mass of $2\ \text{lb}$ experiences a centripetal force of $3.6\ \text{pdl}$ on a path of radius $5\ \text{ft}$. Rearranging the formula gives:

$$v^{2} = \dfrac{F r}{m} = \dfrac{3.6 \times 5}{2} = 9 \quad\rightarrow\quad v = 3\ \text{ft/s} \;\;(\approx 0.914\ \text{m/s})$$

These examples show how the circular motion calculator can solve for any variable when the other three are known.

Energy Conservation and Centripetal Force

Because the centripetal force always acts perpendicular to the direction of motion, it does no work. Hence, no energy is transferred into or out of the system, and the total mechanical energy remains constant. This is why planets and satellites can orbit the Sun or Earth for long periods without slowing down – the gravitational force provides the necessary centripetal pull while leaving the speed unchanged.

Using the Online Centripetal Force Calculator

To use this physics circular motion tool, simply enter the mass, velocity (or angular velocity), and radius. The centripetal force calculator returns the force and acceleration in your chosen units. It also works in reverse: input force and radius to calculate velocity, or set any combination to solve for the missing quantity. The instant update makes it ideal for homework, experiment verification, or engineering design.