Hard

Vertical Circular Motion

Simple Explanation

In a vertical circle, the centripetal force still always points toward the centre — but gravity now points straight down at every point, so it helps supply the centripetal force at the top of the loop and works against it at the bottom, meaning the tension (or normal force) needed is different at every point around the circle.

Why Do We Need It?

This is why a roller coaster or a swung bucket of water needs a minimum speed to complete a vertical loop safely — go too slowly at the top, and there is not enough speed for gravity alone to supply the needed centripetal force, so the object (or the water) falls out of the circular path.

Worked Example

Find the tension at the bottom of a vertical circle

A 2 kg ball on a 1 m string swings in a vertical circle. At the lowest point of the loop, its speed is 5 m/s. Find the tension in the string at that point (g = 9.8 m/s²).

Why Does This Work?

At the bottom of the loop, gravity pulls the ball away from the centre (downward), so tension must supply both the centripetal force AND cancel out gravity's outward pull — making tension at the bottom larger than mv²/r alone. At the top, gravity pulls toward the centre (also downward, which is now inward), so tension only needs to supply the remainder: T = mv²/r − mg. That is also why the minimum speed at the top occurs when tension drops to zero and gravity alone supplies the whole centripetal force.

Real-Life Example

A roller coaster loop

A roller coaster car goes around a vertical loop.

Engineers must ensure the car is moving fast enough at the very top of the loop that gravity alone doesn't exceed the centripetal force needed — otherwise the car (and its passengers) would lose contact with the track and fall.

Practice

A 1.5 kg ball on a 0.8 m string moves at 4 m/s at the bottom of a vertical circle. Find the tension there (g = 9.8 m/s²).

Hard

Common mistake

Using the same equation (T = mv²/r) for every point on a vertical circle — the tension equation actually changes depending on position, because gravity contributes differently at the top, the bottom, and the sides of the loop.

Quick Review

At the bottom of a vertical circle: T = mv²/r + mg.
At the top of a vertical circle: T = mv²/r − mg.
The minimum speed at the top occurs when tension reaches zero and gravity alone supplies Fc.