Skip to lesson

Physics · Class 11

Tangent & inward acceleration

Motion in a Plane

Velocity includes direction. On a circle its magnitude stays constant but its direction continually changes. The velocity is tangent to the circle, while acceleration points inward. The two are perpendicular. Doubling speed quadruples the centripetal acceleration at the same radius.

Open the interactive experiment →

Constant speed can still mean acceleration.

வட்ட இயக்கத்தில் வேகத்தின் அளவு மாறாமல் இருந்தாலும் திசை மாறுகிறது. வேகத் திசையன் தொடுகோட்டுத் திசையில் இருக்கும்; முடுக்கம் மையத்தை நோக்கி இருக்கும். ஆகவே இரண்டும் செங்குத்தானவை.

Key notes, relationships and model limits

Key relationship

ω = ±v/r; θ = θ₀ + ωt; v⃗ = (−rω sinθ, rω cosθ); a⃗ = −ω²r (cosθ, sinθ); |a| = v²/r; T = 2πr/v for v > 0.

Remember

Velocity includes direction. On a circle its magnitude stays constant but its direction continually changes. The velocity is tangent to the circle, while acceleration points inward. The two are perpendicular. Doubling speed quadruples the centripetal acceleration at the same radius.

Common trap

Acceleration is not zero merely because speed is constant. A change in velocity direction also produces acceleration.

Model limits

Fixed circular path, constant nonnegative speed and fixed radius. Velocity is tangent; acceleration points toward the centre. Arrow lengths are normalized independently to make directions clear; magnitudes are in the readouts. Zero speed gives no velocity, acceleration or finite revolution period.

What you will learn

Identify tangent velocity and radial acceleration; relate v, r, ω and period; explain why speed alone is insufficient.

Try the experiment

Play and observe both arrows around the circle. Reverse the direction, then double the speed and compare acceleration.

Think of it this way

A car taking a bend at steady speed still needs an inward acceleration to keep turning.

Worked example

For r=4 m and v=3 m/s: a=9/4=2.25 m/s², |ω|=0.75 rad/s and T=8π/3≈8.378 s. Velocity and acceleration have zero dot product.

Why this matters

An inward force changes the direction of velocity without changing its speed in ideal uniform circular motion.

Before you start

Rectangular axes · Pythagoras · sine and cosine

Check your understanding

At fixed radius, what happens to centripetal acceleration when speed doubles?

Read the explanation

a=v²/r. Replacing v by2v gives a′=(2v)²/r=4a. Reversing travel direction does not reverse the inward acceleration.

Course connection: CBSE XI Motion in a Plane; NEET/JEE kinematics. Use the assumptions stated in each question.