Understanding Circular Motion
Educational Tool
Circular motion is fundamental to understanding everything from car turns to satellite orbits to spinning machinery. This calculator helps you explore the relationships between speed, radius, and the forces required to maintain circular paths.
What Is Circular Motion?
When an object moves in a circle at constant speed, it's undergoing uniform circular motion. Even though the speed stays the same, the velocity is constantly changing because velocity includes direction, and direction is always changing as the object curves.
Any change in velocity requires acceleration. For circular motion, this acceleration always points toward the center of the circle and is called centripetal acceleration.
Centripetal Acceleration
Centripetal means "center-seeking." The acceleration points inward, toward the center of the circular path.
- • a_c: centripetal acceleration (m/s² or ft/s²)
- • v: tangential speed (m/s or ft/s)
- • r: radius of the circle (m or ft)
Notice: mass doesn't appear! Acceleration depends only on speed and radius.
Centripetal Force
From Newton's second law (F = ma), the force needed to produce centripetal acceleration is:
- • F_c: centripetal force (N or lbf)
- • m: mass of the object (kg or lb)
Force DOES depend on mass: heavier objects need more force for the same circular path.
Frequency, Period, and Angular Speed
Frequency (f)
Revolutions per second
Unit: Hz (1/s)
Period (T)
Time for one revolution
T = 1/f
Angular Speed (ω)
Radians per second
ω = 2πf = 2π/T
Connecting to tangential speed: v = ω × r = 2πfr = 2πr/T
How This Calculator Uses the Formulas
You provide at least two known values (like radius and speed, or frequency and radius). The calculator uses the interconnected formulas to derive everything else:
- From v and r → compute a_c = v²/r
- From a_c and m → compute F_c = m × a_c
- From f → compute ω = 2πf and T = 1/f
- From ω and r → compute v = ωr
- And many other combinations...
You can compare multiple scenarios (up to 3 cases) to see how changing parameters affects the required forces and accelerations.
Limitations: Uniform Circular Motion Only
- Constant speed only: This tool assumes the object maintains constant speed around the circle. It does not handle acceleration along the path (tangential acceleration).
- No vertical loops: For vertical circles (like roller coaster loops), gravity complicates the analysis. The force varies around the loop.
- No banking angles: For banked curves (like race tracks), the analysis requires considering the normal force angle. This tool assumes flat circular paths.
- No friction coefficients: While friction often provides centripetal force, this tool doesn't calculate whether available friction is sufficient.
- Educational purposes: This is a learning tool. Real engineering applications require safety factors, material considerations, and professional expertise.
Frequently Asked Questions
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