Centripetal Acceleration Scaling: Doubling Speed Does More Than Doubling Radius

For uniform circular motion, centripetal acceleration has magnitude ac = v²/r and points toward the centre. At fixed radius, doubling speed multiplies this acceleration by four.

Constant speed does not mean zero acceleration

Velocity includes direction. On a circular path its direction changes continuously, so acceleration exists even when speed remains constant. The inward acceleration discussed here changes direction; a separate tangential component appears if the speed also changes.

Before comparing two cases, list what is held fixed. A radius comparison at fixed linear speed differs from a comparison at fixed angular speed. The formulas ac = v²/r and ac = ω²r agree through v = ωr, but they describe different controlled comparisons if different quantities are fixed.

Worked scaling example

A model object moves at 6 m/s on a circle of radius 3 m. Its inward acceleration magnitude is 6²/3 = 12 m/s². Increasing the speed to 12 m/s on that same radius gives 12²/3 = 48 m/s².

The second acceleration is four times the first, because the speed ratio 12/6 = 2 is squared. Now keep speed at 6 m/s and double radius to 6 m. Acceleration becomes 36/6 = 6 m/s², half the original value.

Combine two changes with a ratio

If the new speed is kv and the new radius is hr, then a₂/a₁ = k²/h. This ratio avoids recalculating common factors. For triple speed and double radius, the acceleration ratio is 9/2, not 3/2. Write the square explicitly to prevent that common shortcut error.

At fixed angular speed instead, doubling radius doubles linear speed. Substituting both changes gives acceleration ratio 2²/2 = 2. The apparent reversal is not a contradiction; the phrase “fixed speed” refers to different quantities in the two cases.

Common mistake

Use radius rather than diameter and convert speed units before squaring. A speed in kilometres per hour cannot be squared and combined directly with metres unless the conversion is handled. These ideal circular-motion calculations do not establish safe driving, equipment or amusement-ride limits; real forces and conditions require more information.

Speed ka ratio square hota hai; radius ke ratio se divide karo.

Three practice questions with worked answers

1. Find ac for v = 4 m/s and r = 2 m.

   Use v²/r = 16/2 = 8 m/s², directed inward.

2. At fixed radius, speed becomes half its original value. What is the acceleration ratio?

   The ratio is (1/2)² = 1/4. It is one quarter of the original acceleration.

3. Speed triples and radius also triples. Find a₂/a₁.

   Use k²/h = 3²/3 = 3. The inward acceleration magnitude triples.

Related study guides

• Rotational Motion for JEE: Moment of Inertia Table and Tricks
• Force and Laws of Motion Class 9 Notes with Numericals

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References and authorship

Original examples and practice questions, prepared with AI assistance and editorial arithmetic review. Related definitions and methods: OpenStax: Centripetal Acceleration. No textbook or exam-board affiliation is claimed.