Centripetal Force Calculator
Find the inward force that keeps a mass moving in a circle, from speed, rpm or period, with banked-curve and loop cases.
🎡 What is Centripetal Force?
Centripetal force is the net inward force that makes an object follow a circular path. The word means "center-seeking". Any object moving in a circle at speed v on a radius r is accelerating toward the center at a = v² / r, even if its speed never changes, because the direction of its velocity is always turning. Newton's second law then says the inward force must be F = m v² / r. The same force can be written F = m w² r with the angular velocity w, or F = 4 pi² m r / T² with the period T of one turn. This calculator accepts any of these descriptions and also covers two classic special cases, the frictionless banked curve and the vertical loop.
The key idea is that centripetal force is not a new kind of force. It is a requirement that some real force must meet. For a ball on a string it is the tension, for a car on a flat road it is friction, for a satellite it is gravity, and for an electron in a magnetic field it is the magnetic force. When you draw a free-body diagram you should draw those real forces and then notice that their inward sum equals m v² / r. If the available force falls short, for example on an icy road, the object cannot follow the circle and moves off along a straight tangent. The opposite viewpoint, the outward push felt by someone riding in the rotating frame, is the fictitious force handled by the centrifugal force calculator.
Engineers use these formulas to design highway curves, roller coasters, centrifuges and flywheels. A banked road tilts the normal force so that part of it points toward the center of the curve, and at the right design speed no friction is needed at all. A vertical loop is harder to analyse because gravity helps at the top and opposes at the bottom, so the normal force changes around the circle. Both cases are solved here with the same inertial-frame reasoning. Once you have the force, the force calculator and the net force calculator help you combine it with other forces, and the normal force calculator covers the straight-line cases.
A few common mistakes are worth avoiding. Centripetal force does no work, because it is perpendicular to the velocity, so it never changes the speed. The force grows with the square of the speed, so a small increase in speed demands a large increase in grip. And the radius must be the radius of the path itself, not a diameter or the distance to some other point.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Car on a Flat Curve
A 1200 kg car takes a 50 m radius curve at 72 km/h
Example 2 — A Ball on a String
A 0.5 kg ball swung in a horizontal circle on a 1.2 m string at 2 revolutions per second
Example 3 — A Space Station in Orbit
A 420,000 kg station on a 6,780 km orbital radius with a period of 92.68 minutes
Example 4 — A Banked Highway Curve
A 1200 kg car at 90 km/h on a frictionless curve of radius 120 m
Example 5 — A Roller Coaster Loop
A 70 kg rider on a loop of radius 10 m, passing the top at 12 m/s
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for centripetal force?
Centripetal force is F = m v² / r, where m is the mass, v the speed along the circle and r the radius. It can also be written m w² r or 4 pi² m r / T². A 1200 kg car at 20 m/s on a 50 m radius needs 9600 N.
How do you calculate centripetal force from rpm?
Convert rpm to rad/s with w = rpm x 2 pi / 60, then use F = m w² r. A 0.5 kg ball on a 1.2 m string turning at 2 revolutions per second gives w = 12.566 rad/s and F = 94.75 N.
What provides the centripetal force on a car in a turn?
Friction between the tyres and the road, plus any horizontal component of the normal force on a banked road. If the needed force exceeds what friction can supply, the car slides outward.
What is the ideal banking angle for a curve?
The angle satisfies tan(theta) = v² / (r g). At 25 m/s on a 120 m curve it is 27.97 degrees. At that speed no friction is needed, and the angle does not depend on the mass of the vehicle.
What is the difference between centripetal and centrifugal force?
Centripetal force is the real inward force on an object moving in a circle. Centrifugal force is the apparent outward force felt in the rotating frame. They have the same size, m v² / r, but opposite directions.
What is the minimum speed to complete a vertical loop?
At the top the speed must be at least sqrt(g r), because gravity alone then supplies the centripetal force. At the bottom that corresponds to sqrt(5 g r). For a 10 m radius loop that is 9.90 m/s at the top and 22.14 m/s at the bottom.
How does centripetal force change with radius?
At a fixed speed the force is inversely proportional to the radius, so a tighter turn needs more force. At a fixed rotation rate the force is proportional to the radius instead.
Does centripetal force do work?
No. The force is perpendicular to the velocity at every instant, so it changes the direction of the motion but not the speed or the kinetic energy of the object.
What is centripetal acceleration?
It is the inward acceleration a = v² / r = w² r that points to the centre of the circle. Newton's second law then gives the force as F = m a. At 20 m/s on a 50 m radius it is 8 m/s², or 0.816 g.
Why do you feel pushed outward in a turn?
Your body tends to continue in a straight line, while the car turns toward the centre. From inside the car this looks like an outward push, which is the fictitious centrifugal force of the rotating frame.