Circular motion

Circular motion is a vector problem before it is a formula problem. An object can keep the same speed and still accelerate because its velocity vector keeps turning. The central question is then: what real force, or combination of forces, produces the required inward acceleration?

Constant speed can still mean non-zero acceleration. In uniform circular motion, the magnitude of velocity stays fixed while its direction changes continuously.

Angular and linear descriptions

For motion on a circle of radius r, arc length and angular position satisfy s = rθ when θ is in radians. The angular variables are ω = dθ/dt and α = dω/dt.

Uniform circular motion

v = rω

ω and speed are constant, but velocity direction changes.

Changing speed

a_t = rα = dv/dt

Tangential acceleration changes the magnitude of velocity.

Centripetal acceleration

The inward acceleration required to bend the trajectory has magnitude:

a_c = v²/r = rω² = 4π²r/T²

It points toward the centre of curvature. In uniform circular motion it is perpendicular to v, so it changes direction without changing speed.

The square matters: at fixed radius, doubling speed requires four times the inward acceleration.

Real forces provide the inward net force

Newton’s second law gives F_{inward, net} = ma_c = mv²/r. “Centripetal force” names the inward role of the net force; it is not an additional fundamental force.

Satellite

Gravity provides the inward force.

Car on a level road

Static friction can provide the inward force.

Mass on a string

Tension can provide the inward force.

Worked example: a car on a flat curve

A 1200 kg car travels at 15 m s⁻¹ around a level curve of radius 50 m.

a_c = 15²/50 = 4.5 m s⁻²

F_inward = 1200 × 4.5 = 5.4 × 10³ N

On a level road, static friction must supply this force. If μ_s is the maximum static-friction coefficient, the no-slip condition is approximately v ≤ √(μ_sgr).

Banked curves

Banking tilts the normal force so that it has an inward horizontal component. For an ideal design speed where friction is not required:

tan θ = v²/(rg)

Centripetal force is not an extra force. On an ideal banked turn, the inward component of the normal force provides the required radial net force.

This is a force-balance result, not a universal speed limit. Real tyres, road conditions and vehicle dynamics still matter.

Vertical circles and changing speed

In a vertical circle, the inward direction changes around the path while gravity always points downward. The radial equation must therefore be written separately at the top, side and bottom.

At the top, gravity may help provide the inward force. At the bottom, gravity points away from the centre while the normal force or tension points inward. The required net radial force is still mv²/r.

If speed is changing, total acceleration has radial and tangential components: a_r = v²/r and a_t = dv/dt. Their vector sum gives the actual acceleration.

Exercises

Rotating platform

A point is 0.40 m from the axis of a platform rotating at 3.0 rad s⁻¹. Find its speed and centripetal acceleration.

Solution

v = rω = 1.2 m s⁻¹. a_c = rω² = 3.6 m s⁻².

Speed doubled

A car takes the same curve at twice its original speed. How does the required inward force change?

Solution

At fixed m and r, F ∝ v², so the required inward force becomes four times larger.

Free-body diagram

A satellite moves in a nearly circular orbit. Should you add a separate centripetal-force arrow?

Solution

No. Draw the real force—gravity. Its inward resultant is what produces the centripetal acceleration.