Centripetal Force

The magnitude of the centripetal force can be expressed in terms of linear velocity ($v$) or angular velocity ($\omega$).

Using linear velocity: $F = \frac{mv^2}{r}$ where $m$ is mass, $v$ is tangential speed, and $r$ is the radius of the path.

Using angular velocity: $F = mr\omega^2$ This form is particularly useful when the rotation rate (radians per second) is known.

A third variation relates force to both linear and angular velocity: $F = mv\omega$

In vertical circles, the centripetal force is the resultant of the tension (or normal force) and the object's weight ($mg$).

At the top: Both tension ($T$) and weight act downwards toward the center. Thus, $T + mg = \frac{mv^2}{r}$. The minimum speed to maintain the circle occurs when $T = 0$, leading to $v_{min} = \sqrt{gr}$.

At the bottom: Tension acts upward (toward the center) while weight acts downward (away from the center). Thus, $T - mg = \frac{mv^2}{r}$, meaning tension is at its maximum here: $T = \frac{mv^2}{r} + mg$.

Identify the Source: Always start by identifying which physical force (tension, friction, etc.) is acting as the centripetal force. Never draw 'centripetal force' as an extra arrow on a free-body diagram.

Resultant Force Equation: Set the sum of forces pointing toward the center minus forces pointing away from the center equal to $\frac{mv^2}{r}$.

Check Units: Ensure mass is in kg, velocity in m/s, and radius in meters. If given angular velocity in RPM, convert it to rad/s first.

Sanity Check: In vertical motion problems, the tension at the bottom should always be greater than the tension at the top by at least $2mg$.