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Classical Mechanics

Uniform Circular Motion

Constant speed, changing velocity, and the outward force that does not exist.

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Uniform Circular Motion

Constant speed, changing velocity, and the outward force that does not exist.

18 min2 checkpoints4 simulation tasks

Introduction

Swing a ball on a string in a circle at a steady rate: its speed never changes, yet it is accelerating the whole time.

Predict

You cut the string at the moment the ball is at the top of its circular path. What path does the ball take afterwards?

Let go of the string

Try it

The ball goes round anticlockwise. Choose where to cut the string so the ball flies off straight toward the bottom of the screen.

GoalMake the ball fly off toward the bottom of the screen.

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The string only ever pulls inward, toward your hand. Nothing pulls the ball outward. "Centrifugal force" is how your own inertia feels from a turning frame, and no object supplies it.

How hard the string pulls

The velocity keeps turning toward the centre, so the acceleration points inward, with magnitude

ac=v2r\acc{a_c} = \frac{\vel{v}^2}{r}

By the second law that needs an inward net force F=mv2/rF = mv^2/r. String tension supplies it here; friction does for a car turning a corner, gravity for a satellite.

Try it

The 1 kg ball circles at 6 m/s on a 2 m string, pulling with 18 N. Change only the speed until the tension is 72 N.

GoalMake the string tension 72 N.

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Try it

Back to 6 m/s and 18 N. Now change only the length of the string until the tension halves, to 9 N.

GoalMake the string tension 9 N.

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Practice

A kg ball on a string m long moves in a circle at m/s. What is the tension in the string?

N

Constant speed, changing velocity

Read the graph

Run the ball at 6 m/s on its 2 m string. Its speed never changes. What is the largest value the axa_x graph reaches?

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m/s²

Run the simulation first, then read the value off it.

Practice

The speed graph for uniform circular motion is a flat line. What does that tell you about the acceleration?

What comes next

You let a ball fly off along its tangent and felt the string's pull grow with the square of the speed. Tracking forces through every instant works, but it is laborious; next comes a shortcut that answers questions about the end of a journey without knowing anything about the middle.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “A released object keeps curving”
  • “Centrifugal force pushes the object outward”
  • “Constant speed means no acceleration”

Finished with this lesson?

Progress is stored on this device only. There is no account and nothing is sent anywhere.

Research behind this lesson

  • McCloskey et al. 1980: Asked what path a ball takes on leaving a curved tube, about half of college students, including some who had studied physics, drew a curving path, as though the object retained a memory of its circular motion; about a third did the same for a ball swung on a string.
  • Viennot 1979: Students spontaneously supply a force in the direction of motion and treat it as proportional to the speed: a pattern that appears across countries, languages and curricula.
  • Hestenes et al. 1992: A 29-item instrument, expanded to 30 in the 1995 revision, whose wrong answers are all documented student beliefs. It exposed how little conceptual understanding traditional courses were producing, and reoriented a generation of physics teaching.