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Home/Classical Mechanics/Circular Motion

Circular Motion

A compact model of uniform circular motion: speed stays constant in magnitude, but the velocity vector keeps turning. That requires centripetal acceleration a_c = v²/r = ω²r and a net force F = ma toward the center. Adjust radius, mass, and angular speed to see how velocity, acceleration, and required force change.

Who it's for: High school and introductory college physics students studying dynamics, forces, and uniform circular motion.

Key terms

  • Centripetal Force
  • Centripetal Acceleration
  • Uniform Circular Motion
  • Tangential Velocity
  • Angular Velocity
  • Period
  • Radius of Curvature
  • Newton's Second Law

Live graphs

Motion

2.2 m
1.2 rad/s
1 kg

Top view: θ increases so the object moves counterclockwise. Speed v = ωr, centripetal acceleration a = ω²r toward the center.

Vectors

Shortcuts

  • •Space or Enter — start rotation
  • •R — reset

Measured values

v = ωr2.640m/s
a_c = ω²r3.168m/s²
F_c = mω²r3.17N
Position (x,y)2.20, 0.00m
Velocity (v_x,v_y)0.00, -2.64m/s
t0.00s

How it works

In uniform circular motion the speed is constant but the velocity vector turns, so there is a centripetal acceleration toward the center of the circle: a = ω²r = v²/r. The net force required is F = ma, pointing in the same direction as the acceleration (inward). Tension, gravity, or normal force can provide this in real setups.

Key equations

v = ω r,   T = 2π/ω
a = ω²r = v²/r  (toward center)
F = m a  (centripetal force)

Frequently asked questions

If the force is always inward, why doesn't the object spiral into the center?
The centripetal force is perpendicular to the object's velocity. In uniform circular motion, this force changes only the direction of the velocity vector, not its magnitude (speed). It constantly pulls the object away from its straight-line inertial path, bending it into a circle without doing work to pull it radially inward. If the force were removed, the object would fly off tangentially, not spiral inward.
Is centripetal force a new, separate type of force like gravity or friction?
No. Centripetal force is not a new force; it is a descriptive label for the net force component directed toward the center of a circular path. This net force can be supplied by tension (as in this simulator), gravity (planetary orbits), friction (a car turning a corner), or a normal force. It is the role the force plays, not its origin.
What happens if I increase the speed while keeping the radius constant?
The required centripetal force increases with the square of the speed (F_c = m v^2 / r). A small increase in speed requires a much larger tension in the string. In a real system, this could cause the string to break if it exceeds its tensile strength, demonstrating why sharp, high-speed turns require substantial forces.
Does this model apply to vertical circular motion, like a roller coaster loop?
This specific simulator models horizontal motion where the force magnitude is constant. Vertical circular motion is more complex because gravity's direction relative to the path changes. The centripetal force requirement (m v^2 / r) still holds at every point, but the net force providing it (e.g., tension + gravity) varies in magnitude, causing the speed to change if non-conservative forces are absent.