Centripetal Force Formula (Fc) with Examples

Meaning: Centripetal force is the inward (center-seeking) net force that keeps an object moving in a circular path. No inward force = object goes straight along the tangent (Newton said "no curve without a cause").

Main Formula

Fc = (m × v²) ÷ r

What Each Symbol Means

  • Fc = centripetal force (unit: Newton, N)
  • m = mass (kg)
  • v = speed along the circle (m/s)
  • r = radius of the circular path (m)

Why This Formula Works

The centripetal acceleration in circular motion is: ac = v² ÷ r Using Newton's 2nd law F = m × a, we get: Fc = m × (v² ÷ r) So the final formula is: Fc = (m × v²) ÷ r

Direction of Centripetal Force

  • Centripetal force always acts toward the center of the circle.
  • Velocity is tangential, but force is inward, so the object keeps turning instead of escaping like it has exam fear.

Alternate Forms

If you know angular speed (ω) instead of linear speed (v):
  • v = ω × r
  • ac = r × ω²
  • Fc = m × r × ω²

Quick Unit Check

  • kg × (m²/s²) ÷ m = kg·m/s² = N

3 Sample Questions with Solutions

Question 1: Find Centripetal Force

A 1.2 kg ball is tied to a string and rotated in a circle of radius 0.8 m at a speed of 6 m/s. Find the centripetal force. Solution Given: m = 1.2 kg, v = 6 m/s, r = 0.8 m Formula: Fc = (m × v²) ÷ r Fc = (1.2 × 6²) ÷ 0.8 Fc = (1.2 × 36) ÷ 0.8 = 43.2 ÷ 0.8 Fc = 54 N

Question 2: Find Speed (v) from Force

A car of mass 900 kg takes a circular turn of radius 50 m. The required centripetal force is 8100 N. Find the speed of the car. Solution Given: Fc = 8100 N, m = 900 kg, r = 50 m Formula: Fc = (m × v²) ÷ r v² = (Fc × r) ÷ m v² = (8100 × 50) ÷ 900 v² = 405000 ÷ 900 = 450 v = √450 ≈ 21.21 m/s

Question 3: Using Angular Speed (ω)

A 0.5 kg object rotates in a circle of radius 0.4 m with angular speed ω = 10 rad/s. Find the centripetal force. Solution Given: m = 0.5 kg, r = 0.4 m, ω = 10 rad/s Formula: Fc = m × r × ω² Fc = 0.5 × 0.4 × 10² Fc = 0.5 × 0.4 × 100 = 0.5 × 40 Fc = 20 N

Exam-Trick Notes

  • If v doubles, then Fc becomes 4× (because of v²).
  • If r increases (bigger circle), needed Fc decreases for the same speed.
  • Centripetal force is not a "new" force. It can be provided by tension, friction, gravity, or normal force, as long as the net force is inward.

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