Tension as Centripetal Force
Simple Explanation
When an object is swung in a circle on a string, the tension in the string is what pulls it toward the centre β the string is the real, physical force supplying the centripetal force.
Why Do We Need It?
This is the clearest, simplest example of a centripetal force in action: you can feel the tension yourself when you swing something on a string, and it disappears the instant the string is cut β proving the object was never being pushed outward at all.
See It
A diagram showing a ball on a circular path with a red arrow labelled Tension pointing inward along the string toward the centre
Formula
Centripetal Force (from angular velocity)
Fc = mrΟΒ²
The same centripetal force, written using angular velocity instead of linear speed β useful whenever rotation rate (rad/s) is what you are given.
- Fc
- β centripetal force, in newtons (N)
- m
- β mass of the object, in kilograms (kg)
- r
- β radius of the circular path, in metres (m)
- Ο
- β angular velocity, in radians per second (rad/s)
When to use it: Whenever you know how fast something is rotating in radians per second (rather than its linear speed) β substitute v = rΟ into Fc = mvΒ²/r to get this form.
Worked Example
Find the tension in a swung string
A 0.2 kg ball on a 0.5 m string swings in a horizontal circle at an angular velocity of 5 rad/s. Find the tension in the string.
Why Does This Work?
A string can only pull, not push, and it can only pull along its own length β straight from the object back toward whatever it is anchored to. For an object swinging in a horizontal circle, that anchor is the centre, so the string's pull is automatically the centripetal force.
Real-Life Example
A hammer throw in athletics
An athlete swings a heavy ball on a wire before releasing it.
The wire supplies the centripetal force that keeps the ball moving in a circle β the moment the athlete lets go, the wire's tension vanishes and the ball flies off in a straight line, tangent to the circle.
Practice
A 0.3 kg ball on a 0.4 m string swings at 4 rad/s. What is the tension in the string?
MediumCommon mistake
Assuming the string also has some outward tension balancing the inward pull β a string only ever pulls in one direction (toward its anchor), so there is nothing pulling the object outward.
Quick Review
- For an object on a string in a horizontal circle, tension is the centripetal force.
- T = mrΟΒ² (or equivalently T = mvΒ²/r).
- Cutting the string removes the centripetal force, and the object flies off in a straight line.