Circular Motion Calculator

Uniform Circular Motion

Uniform Circular Motion Formulas:

Centripetal Acceleration: a_c = v²/r = ω²r
Centripetal Force: F_c = ma_c = mv²/r = mω²r
Angular Velocity: ω = v/r = 2π/T = 2πf
Linear Velocity: v = ωr = 2πr/T
Period: T = 2π/ω = 2πr/v
Frequency: f = 1/T = ω/(2π)

Radius (r) - meters:

Linear Velocity (v) - m/s:

Angular Velocity (ω) - rad/s:

Mass (m) - kg:

Period (T) - seconds:

Frequency (f) - Hz:

Centripetal Acceleration (a_c) - m/s²:

Centripetal Force (F_c) - N:

Banked Curves and Friction

Banked Curve Formulas:

No Friction: tan(θ) = v²/(rg)
With Friction: F_friction = μN
Normal Force on Banked Surface: N = mg cos(θ)
Maximum Speed: v_max = √[rg(μ cos(θ) + sin(θ))/(cos(θ) - μ sin(θ))]

Banking Angle (θ) - degrees:

Coefficient of Friction (μ):

Radius (r) - meters:

Mass (m) - kg:

Velocity (v) - m/s:

Gravity (g) - m/s²:

Typical Friction Coefficients:
Ice: μ ≈ 0.1
Wet Road: μ ≈ 0.4
Dry Road: μ ≈ 0.7
Racing Tires: μ ≈ 1.0+

Vertical Circular Motion

Vertical Circle Formulas:

At Top: T_top + mg = mv²/r → T_top = mv²/r - mg
At Bottom: T_bottom - mg = mv²/r → T_bottom = mv²/r + mg
Minimum Speed at Top: v_min = √(gr)
Energy Conservation: ½mv_bottom² = ½mv_top² + mg(2r)

Radius (r) - meters:

Mass (m) - kg:

Speed at Bottom (v_bottom) - m/s:

Speed at Top (v_top) - m/s:

Tension at Bottom (T_bottom) - N:

Tension at Top (T_top) - N:

Critical Conditions:
At top: T_top ≥ 0 for complete loop
Minimum speed: v_top ≥ √(gr)
Loop-the-loop rides use this principle

Circular Motion Concepts:

Centripetal Force: Always directed toward the center of the circle, provides the acceleration needed for circular motion.

Banking: Tilting the road surface to help provide centripetal force and reduce dependence on friction.

Vertical Loops: Require minimum speeds to maintain contact with the track throughout the motion.

Angular vs Linear: Angular quantities (ω, α) describe rotation; linear quantities (v, a) describe motion along the path.