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All lessons Mechanics22 min

Circular Motion & Centripetal Force

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01
Hook
02
Explore
03
Formalize
04
Practice
05
Challenge
Interactive simulation
01

Hook

Swing a bucket of water in a vertical circle over your head and at the top — bucket upside down — the water stays in. Slow down a little and you get soaked. What holds the water in, and how slow is too slow?

02

Explore

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03

Formalize

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Practice

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Challenge

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Spoilers

Circular Motion & Centripetal Force — summary and key formula

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The question

Swing a bucket of water in a vertical circle over your head and at the top — bucket upside down — the water stays in. Slow down a little and you get soaked. What holds the water in, and how slow is too slow?

Nothing pushes the water outward; the bucket is yanked inward faster than gravity can pull the water out, so the bottom keeps catching the falling water. Below a critical speed gravity wins — the same inward 'centripetal' force that bends every roller-coaster loop, satellite orbit, and highway curve.

The key idea

An object moving in a circle is always accelerating toward the centre — even at constant speed — because its velocity DIRECTION keeps changing, and any change in velocity needs a force. This centripetal ('centre-seeking') force points inward, perpendicular to the velocity.

As the velocity vector rotates while keeping length v, its inward change Δv works out to ac = v²/r toward the centre; times mass gives Fc = mv²/r. The simulation makes this visible: double v and Fc jumps ×4, double m and it doubles, halve r and it doubles. Centripetal force is never a new force — it's SUPPLIED by something familiar: tension (ball on a string), gravity (a satellite), friction (a car on a bend), or the normal force (a loop or bucket rim). Identifying the supplier is the key to every problem. At the top of a vertical loop, gravity can supply at most mg of inward force, setting the minimum speed v_min = √(gr) for the cart — or the water — to stay in contact. **Connect it:** ac=v2/ra_c = v^2/rac​=v2/r is not a new force law — it is what Newton's second law demands when a velocity vector changes direction at constant magnitude; some real force (tension, gravity, friction, the road) must be drafted to supply mv2/rmv^2/rmv2/r, and if none can, the circle ends and the body departs along the tangent.

The formula

Fc=mv2rac=v2rF_c = \dfrac{mv^2}{r} \qquad a_c = \dfrac{v^2}{r}Fc​=rmv2​ac​=rv2​
  • ·Fc = centripetal force (N)
  • ·ac = centripetal acceleration (m/s²)
  • ·m = mass (kg)
  • ·v = speed (m/s)
  • ·r = radius (m). Since Fc = m·ac
  • ·the two are one statement written twice. Note v is SQUARED while m and r are linear.

Common mistake

Thinking a force flings you OUTWARD — the net force actually points INWARD (centripetal); you feel pushed out only because your inertia wants to go straight.

What to remember

  • ·Circular motion needs an inward (centripetal) net force, Fc = mv²/r.
  • ·Velocity is tangent (sideways); the force is perpendicular, toward the centre.
  • ·Force grows with the SQUARE of speed and inversely with radius.