TwiddleLab / Physics / Gravity and Orbits

Moving in Circles: a = v²/r

Lesson 2 of 10 · 7 min · Free, no account needed

What you will learn

• Going round at a steady speed is still accelerating: the velocity keeps turning, and the change points at the centre.
• a = v² / r: twice the speed gives four times the acceleration, twice the radius gives half.
• One lap takes T = 2πr / v.

Start the lesson

How the lesson goes

1. Take a highway exit twice as fast. Does the push feel twice as strong, or more?

2. Steady speed, but is the velocity steady? Snap two close moments and compare.

3. How big is that acceleration? Solve three dial puzzles.

   The mint arrow is the acceleration. In each puzzle one dial is locked: use the other one.

4. Centripetal acceleration

   Moving round a circle at speed v, an object accelerates toward the centre with a = v² / r. This is centripetal acceleration.

   Something must pull it inward: tyre grip for a car, a string for a ball, gravity for the Moon. Newton's second law gives the force: F = m v² / r.

   One lap is the distance 2πr, so the time for a lap is T = 2πr / v.

5. Design the exit: the tightest curve a car at 20 m/s can take safely.

   Drag the yellow handle. The tyres give at most 7 m/s² of sideways grip.

   The car needs a = v² / r = 400 / r toward the centre, and only grip can supply it. That stays under 7 m/s² when r is at least 400 / 7, about 57 m.

6. Game: Centrifuge trainer. Spin the pilot to 3 g, 5 g and 8 g.

   Pilots train in a cabin on an 8 m arm. One spin per round.

7. Math: one of these graphs is a against v, the other a against r. Which is a against v?

   Drag the dot on each graph. The yellow dot shows the input doubled.

   With r fixed, a = v² / r is a square function of v, a parabola. With v fixed, a = v² / r is a constant over r, a hyperbola: double r and a halves.

8. A car takes a curve of radius 50 m at 20 m/s. What is its acceleration?

   a = v² / r = 20 × 20 / 50 = 8 m/s².

9. Your centrifuge rounds. Tap your best one.

Previous lesson: Newton's Cannon: Falling Around the Earth · Next lesson: Universal Gravitation: Inverse Square · Part of the course: Gravity and Orbits

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