Centripetal Force

What keeps anything moving in a circle — and what happens the instant it stops. Swing a mass on a string and watch the one force that matters: the centripetal force, always pointing to the centre, with magnitude F = m·v²/r. Set the mass, radius and speed and read the force, acceleration, angular speed and period live, with the velocity arrow (tangent) and force arrow (inward) drawn each frame. Then cut the string and see the mass fly off along the tangent — not straight outward. It all runs on your device.

You are in the Physics lab.

Blue = velocity (tangent) · Red = centripetal force (inward). Hit Release to cut the string.
Presets
Speed v
Angular speed ω
Centripetal force Fc
Centripetal accel ac
Period T
Fc = m·v²/r grows with the square of speed — the dot marks your current setting.
The force points to the centre — always. Circular motion needs a constant inward pull: F = m·v²/r. Double the speed and the force quadruples; halve the radius and it doubles. The velocity, meanwhile, points along the tangent, at right angles to the force — which is why the moment you cut the string the mass leaves along that tangent in a straight line, not flung outward along the radius. That right angle between v and F is the whole idea of circular motion.

Reading the simulation

What the arrows, the graph and the readouts are telling you.

1

The force and the velocity are perpendicular

The red arrow is the centripetal force, drawn along the radius line and always pointing to the centre; the blue arrow is the velocity, along the tangent. They sit at a right angle to each other the whole way round. The force keeps changing the direction of the velocity without changing its size, which is exactly what "moving in a circle at constant speed" means. The radius line is labelled with r so you can see the geometry the formula uses.

2

The graph shows Fc = m·v²/r

The companion plot draws centripetal force against speed for your current mass and radius, and the blue dot marks where you are on it. Because v is squared the curve steepens fast: slide the speed up and the dot climbs the steep part — doubling v quadruples F. Shrinking the radius or raising the mass lifts the whole curve, since F grows as 1/r and as m. The five readouts — v, ω, Fc, ac and T — update on every frame to match.

3

Release → a straight tangent line

Press Cut the string (or the Release button) and the inward force vanishes. With no net force Newton's first law takes over and the mass travels in a straight line at constant velocity — along the tangent it had at the instant of release. It does not shoot outward along the radius. Watching that tangential flyoff is the fastest cure for the most common misconception about circular motion.

How It Works

One inward force, a right angle, and Newton's first law waiting in the wings.

1

The inward force

To move in a circle at constant speed, an object must accelerate — not because its speed changes, but because its direction does. That acceleration points to the centre, a = v²/r, so by Newton's second law there must be a net inward force F = m·v²/r. It is always some real force in disguise: string tension here, gravity for a satellite, friction for a car cornering.

2

Speed, radius, mass

Because v is squared, speed dominates: going twice as fast needs four times the force. A tighter circle (smaller r) also needs more force, and a heavier mass needs proportionally more. The angular speed ω = v/r and period T = 2π/ω tell you how quickly it goes around. Slide the three controls and watch all four readouts respond.

3

Cut the string

The velocity is tangent to the circle, perpendicular to the force. Remove the force — press Release — and Newton's first law takes over: no net force means straight-line motion at constant velocity, along that tangent. It does not shoot outward along the radius. Seeing the tangential flyoff is the fastest way to fix the most common misconception about circular motion.

What is centripetal force?
Centripetal force is the net inward force that keeps an object moving in a circle. It always points toward the centre and has magnitude F = m·v²/r. It is not a new kind of force — it is whatever real force plays that role: the tension in a string, gravity for an orbit, friction for a car turning, or the normal force on a banked track. This simulator draws that inward force as a red arrow while the mass circles.
What is the formula for centripetal force and acceleration?
The centripetal acceleration is a = v²/r = ω²·r, directed toward the centre, and by Newton's second law the force is F = m·a = m·v²/r. The angular speed is ω = v/r and the period is T = 2π/ω = 2πr/v. Set the mass, radius and speed and the tool shows all five live and plots Fc against speed with your operating point marked.
Why does the object fly off along the tangent when the string is cut?
Because the centripetal force is the only thing bending the path into a circle. Remove it and Newton's first law takes over: with no net force the object continues in a straight line at constant velocity, tangent to the circle at the moment of release. It does not fly straight outward along the radius — the most common misconception. Press Release to see it.
Is centripetal force the same as centrifugal force?
No. The centripetal force is a real inward force in an inertial frame. The centrifugal force is an apparent outward force that only appears in a rotating reference frame — a bookkeeping term, not a push from any object. In the ground frame this simulator uses, there is only the inward centripetal force; the outward feeling a passenger reports is their inertia resisting the turn.

Teaching circular motion? You can embed this simulator on your own site free — one line of code, no sign-up.