Centripetal Force Calculator

Solve circular-motion problems using \(F_c = \dfrac{m v^2}{r}\). Choose a target, enter the known values, and get force, acceleration, angular speed, and period instantly.

Configuration

Pick what to solve for and your preferred output units.

Inputs

Result

Practical Guide

Centripetal Force Calculator: How to Use It and Interpret Results

A field-ready walkthrough for circular-motion problems. Learn what the centripetal force equation assumes, how to pick the right “Solve For” target, and how to sanity-check outputs like acceleration, angular speed, and period.

6–8 min read Updated 2025 Mechanical / Physics

Quick Start

The calculator is built around the standard centripetal force relationship for uniform circular motion: \[ F_c = \frac{m v^2}{r} \] where the net force points toward the center of the circle. Follow these steps to avoid the most common input mistakes.

  1. 1 Choose what you want to solve for: force, mass, speed, or radius. The calculator hides the solved variable automatically.
  2. 2 Enter the known values in the visible fields. Double-check that each one is positive and physically reasonable (no negative radii or speeds).
  3. 3 Pick units next to each input (kg/lbm, m/s-mph, m-ft, N-lbf). The tool converts everything internally to SI before computing.
  4. 4 If you’re solving for force, confirm that the speed you entered is tangential speed (along the path), not angular speed.
  5. 5 Read the main result, then scan Quick Stats for derived quantities: \[ a_c = \frac{v^2}{r},\quad \omega = \frac{v}{r},\quad T=\frac{2\pi}{\omega},\quad f=\frac{1}{T} \]
  6. 6 If the number surprises you, do a quick sensitivity check: change only one variable (like radius or speed) and observe the direction and magnitude of change.
  7. 7 For design work, compare the centripetal force to available friction, tension, or structural capacity to ensure the motion is actually possible and safe.

Tip: If you know rotational rate in RPM instead of speed, convert first: \[ \omega = \frac{2\pi\,\text{RPM}}{60},\quad v=\omega r \] Then use \(v\) in the calculator.

Watch your radius: Use the distance from the path’s center to the object’s center of mass. Using diameter by accident is the #1 reason people get a force that’s 2× off.

Choosing Your Method

In real problems, you often have different “knowns.” The calculator supports the four most common rearrangements of the centripetal force equation. Pick the solve target that matches your data and design intent.

Method A — Solve for Centripetal Force

Use when you know mass, speed, and radius and want the inward force demand.

  • Standard for vehicle curves, rotating machinery, lab demos, and roller-coaster checks.
  • Directly shows load a track, cable, or structure must supply.
  • Best starting point for safety and capacity comparisons.
  • If speed is uncertain, the force can be very wrong (because of \(v^2\)).
  • Assumes constant speed and circular path.
\[ F_c = \frac{m v^2}{r} \]

Method B — Solve for Speed

Use when force capacity is known (e.g., maximum friction or tension) and you need the safe speed.

  • Great for “how fast can we go?” design questions.
  • Maps directly to constraints like tire friction, belt tension, or bearing limits.
  • Easy to pair with factors of safety.
  • Requires a reliable estimate of available inward force.
  • Still assumes uniform circular motion.
\[ v = \sqrt{\frac{F_c r}{m}} \]

Method C — Solve for Radius or Mass

Use radius for geometry/layout design and mass for payload sizing.

  • Radius solve is common for roadway/track layout: “How tight can a curve be?”
  • Mass solve helps for rotating systems with fixed force limits.
  • Useful for early feasibility checks.
  • Be sure your force input is the net inward force, not just one component.
\[ r=\frac{m v^2}{F_c},\quad m=\frac{F_c r}{v^2} \]

What Moves the Number the Most

Centripetal force is a simple equation, but it’s easy to underestimate how sensitive it is to certain inputs. These are the dominant levers.

Speed \(v\) (squared)

Force scales with \(v^2\). Doubling speed makes centripetal force 4× larger. Small speed errors create big force errors.

Radius \(r\) (inverse)

Force scales with \(1/r\). Halving the turn radius doubles force demand. Tight curves are force multipliers.

Mass \(m\) (linear)

Force scales directly with mass. Payload changes or density variations translate 1:1 to force changes.

Net inward force vs. components

The equation requires the resultant inward force. If your inward force comes from tension plus friction or banking, combine them correctly first.

Uniform motion assumption

The calculator assumes constant speed on a circular path. If speed changes or the path isn’t a true circle, treat results as local/instantaneous estimates.

Unit conversions

Mixed units (mph with feet, lbm with lbf) are fine only if you set each unit selector correctly. One wrong unit can shift the result by an order of magnitude.

Rule of thumb: if your speed uncertainty is ±10%, your force uncertainty is roughly ±20% because of the square term.

Worked Examples

Example 1 — Force on a Turning Vehicle

  • Scenario: A 1,400 kg car takes a flat curve at 20 m/s (~45 mph).
  • Curve radius: 60 m.
  • Goal: Find required centripetal force and acceleration.
1
Use core equation: \[ F_c=\frac{m v^2}{r} \]
2
Substitute values: \[ F_c=\frac{(1400)(20^2)}{60} \]
3
Compute: \[ F_c=\frac{(1400)(400)}{60}=9333\ \text{N}\ (\approx 9.33\ \text{kN}) \]
4
Centripetal acceleration: \[ a_c=\frac{v^2}{r}=\frac{400}{60}=6.67\ \text{m/s}^2\ (\approx 0.68g) \]

Interpretation: The tires and road must supply about 9.3 kN of net inward force. If friction is the only source, the required friction coefficient is roughly \[ \mu \approx \frac{a_c}{g}=\frac{6.67}{9.81}=0.68 \] which is high but possible on dry pavement. A smaller radius or higher speed would push beyond typical tire grip.

Example 2 — Safe Speed for a Rotating Mass on a Cable

  • Scenario: A 2.5 kg mass is spun in a horizontal circle on a light cable.
  • Radius: 0.9 m.
  • Max allowable tension (net inward force): 120 N.
  • Goal: Find maximum safe tangential speed.
1
Rearrange for speed: \[ v=\sqrt{\frac{F_c r}{m}} \]
2
Substitute: \[ v=\sqrt{\frac{(120)(0.9)}{2.5}} \]
3
Compute: \[ v=\sqrt{\frac{108}{2.5}}=\sqrt{43.2}=6.57\ \text{m/s} \]
4
Check angular speed: \[ \omega=\frac{v}{r}=\frac{6.57}{0.9}=7.30\ \text{rad/s} \]

Interpretation: At about 6.6 m/s, the cable reaches its inward-force limit. Because force grows with \(v^2\), a small speed increase (say to 7.2 m/s) would raise tension to: \[ F_c=\frac{m v^2}{r}=\frac{2.5(7.2^2)}{0.9}=144\ \text{N} \] which exceeds the allowable load.

Common Layouts & Variations

The same centripetal force model shows up in many engineering contexts. The table below summarizes typical configurations, what usually supplies the inward force, and what you should watch for.

Configuration / Use CaseInward Force SourceWhy It MattersCommon Pitfalls
Flat vehicle curveTire-road frictionSets max speed before skidUsing diameter instead of radius; ignoring wet/ice friction drop
Banked curve / trackNormal force component + frictionBanking reduces friction demandForgetting to resolve forces to get net inward \(F_c\)
Turntable or rotorStructural restraint / bearing reactionControls stress, bearing sizingUsing RPM directly as \(v\); ignoring eccentric mass
Mass on string (lab)TensionTension limit gives safe speedNot accounting for string angle if not horizontal
Roller coaster loopTrack normal force + weight componentNormal force drives rider G-loadsAssuming constant speed through loop
Pipe bend / fluid turnPressure + momentum changeCreates bend thrust loadsUsing particle centripetal model without momentum balance
  • Confirm whether speed is constant or changing.
  • Use net inward force, not a single force component.
  • Check geometry: radius to the center of path.
  • For rotating parts, verify mass location (eccentricity).
  • Account for environment (wet friction, temperature, wear).
  • Compare outputs to practical benchmarks (e.g., g-levels for people).

Specs, Logistics & Sanity Checks

Because this calculator is simple, it’s most powerful as a fast decision tool. Use these checks to ensure the number you’re about to design around is meaningful.

Assumptions to Verify

  • Motion is circular with constant radius.
  • Speed is approximately constant (uniform motion).
  • The inward force you’re using is the net inward force.
  • Mass is concentrated (point-mass approximation) or equivalent mass is justified.

Practical Benchmarks

Quick-stats help interpretation:

  • Acceleration: \(a_c/g\) gives G-level. Humans feel sustained loads above ~2–3g strongly.
  • Angular speed: Convert to RPM if needed: \(\text{RPM}=\omega\cdot 60 /(2\pi)\).
  • Period: If \(T\) seems too small/large, recheck speed units.

Design Notes

  • For friction-limited systems, compare \(F_c\) to available friction \(F_f=\mu N\).
  • For tension-limited systems, compare \(F_c\) to rated tension with a factor of safety.
  • If radius is variable, design for the minimum expected radius.

Dynamic cases: If speed changes significantly (spirals, entry/exit of a curve, non-circular paths), the true inward force is not constant; use this tool for instantaneous checks and follow with a full dynamics model.

Sensitivity trick: If you’re unsure about speed, compute two cases: \(v_{low}\) and \(v_{high}\). The force ratio is \((v_{high}/v_{low})^2\). This gives an immediate design envelope.

Frequently Asked Questions

What is centripetal force in simple terms?
Centripetal force is the net inward force that keeps an object moving in a circle. It is not a new type of force by itself—it can be provided by tension, friction, gravity, normal force components, or a combination of these. The required magnitude for uniform circular motion is \(F_c = m v^2 / r\).
Why does centripetal force depend on speed squared?
In circular motion, the direction of velocity changes continuously. The faster the object moves, the more rapidly its direction must change, which requires a larger acceleration toward the center. That acceleration is \(a_c = v^2/r\), so the force \(F_c = m a_c\) inherits the \(v^2\) term. Doubling speed increases force by a factor of four.
Can I use RPM instead of speed?
Yes, but convert to angular speed first: \(\omega = 2\pi\,\text{RPM}/60\). Then compute tangential speed with \(v=\omega r\) and enter \(v\) into the calculator. Alternatively, compute force directly as \(F_c = m\omega^2 r\) if you prefer working in angular terms.
What’s the difference between centripetal and centrifugal force?
Centripetal force is the real inward force required for circular motion. “Centrifugal force” is an apparent outward force observed in a rotating reference frame; it’s not a real interaction in an inertial frame. For engineering calculations in stationary frames, use centripetal force.
How do I know if my result is reasonable?
Start with acceleration: compute \(a_c = v^2/r\) and compare to \(g\). If you get several g’s for a passenger vehicle on a normal road curve, something is off (often the radius or speed units). Also check that your radius is not accidentally the diameter.
Does the calculator work for non-uniform or elliptical motion?
It is accurate for uniform circular motion (constant speed and constant radius). For non-uniform motion, it still gives the instantaneous inward force demand at that moment, but total dynamics may include tangential acceleration and changing radius. Use it as a quick check, then model the full path if needed.
What produces centripetal force in real systems?
Any force component pointing toward the center can supply centripetal force: cable tension, tire friction, gravitational attraction for orbits, or normal forces from guides/tracks. The calculator tells you how large the net inward requirement is—you still need to verify the physical mechanism can supply it safely.
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