Centripetal Force Calculator
Solve for centripetal force, mass, velocity, or radius for circular motion. Supports linear velocity, angular velocity, frequency, and period. Includes 5 real-world presets, g-force output, and circular motion diagram.
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Velocity input type
What Is the Centripetal Force Calculator?
Centripetal force is the net inward force that keeps an object moving in a circular path. Without it, the object would fly off in a straight line, Newton's first law at work. The word "centripetal" comes from Latin meaning "seeking the centre": the force always points toward the axis of rotation, never along the direction of motion.
Critically, centripetal force is not a new type of force. It is whatever real force, gravity, tension, friction, normal force, or a combination, happens to be directed toward the centre. A satellite stays in orbit because gravity is the centripetal force. A car corners because tyre friction acts toward the centre of the turn. A ball on a string swings because string tension pulls inward.
This calculator lets you solve for any one unknown (force, mass, velocity, or radius) given the other three, across four ways to express rotational speed: linear velocity, angular velocity, frequency, or period.
Formula
Primary Formula, Centripetal Force
F = m × v² / r
where:
F = centripetal force (N), directed toward the centre of rotation
m = mass of the object (kg)
v = linear (tangential) speed (m/s)
r = radius of the circular path (m)
Equivalent Forms Using Angular Velocity & Frequency
F = m × ω² × r (ω = angular velocity in rad/s)
F = m × (2πf)² × r (f = frequency in Hz)
F = m × (2π/T)² × r (T = period in seconds)
Conversion identities:
v = ω × r ω = 2πf = 2π/T
f = 1/T T = 2π/ω
Centripetal Acceleration & G-Force
a_c = v² / r = ω² × r = F / m
g-force = a_c / 9.81 (dimensionless multiple of Earth gravity)
Velocity-Type Relationships
| Quantity | Symbol | Unit | Relationship |
|---|---|---|---|
| Linear velocity | v | m/s | Arc length covered per second along the circular path |
| Angular velocity | ω | rad/s | Angle swept per second; v = ωr |
| Frequency | f | Hz | Revolutions per second; ω = 2πf |
| Period | T | s | Time for one full revolution; T = 1/f = 2π/ω |
How to Use
- 1Choose what to solve for: Click one of the four Solve For buttons, F (force), m (mass), v (velocity), or r (radius). The corresponding input field disappears since it is the unknown.
- 2Select a velocity type: Pick how you know the rotational speed: linear velocity v (m/s, km/h, mph), angular velocity ω (rad/s, rpm, °/s), frequency f (Hz, mHz), or period T (ms, s, min, h).
- 3Enter known values: Type mass, velocity/speed, and radius into the input fields. Use the unit dropdowns on each field to match your data.
- 4Try a preset (optional): Click Car Cornering, Roller Coaster, Washing Machine, Cyclist, or ISS Orbit to pre-fill realistic values and see how the calculator works.
- 5Calculate: Press Enter or click "Calculate". The result panel shows centripetal force, mass, linear velocity, radius, angular velocity ω, frequency f, period T, centripetal acceleration a_c, and g-force, all at once.
Example Calculation
A 1,200 kg car navigates a roundabout with a radius of 30 m at a speed of 40 km/h. What centripetal force does the road need to provide, and how many g does the driver feel?
Given:
m = 1,200 kg
v = 40 km/h = 40 / 3.6 = 11.11 m/s
r = 30 m
Step 1, Centripetal force:
F = m × v² / r
F = 1200 × 11.11² / 30
F = 1200 × 123.43 / 30 = 4,937 N ≈ 4.94 kN
Step 2, Centripetal acceleration:
a_c = v² / r = 11.11² / 30 = 4.11 m/s²
Step 3, G-force on driver:
g-force = a_c / 9.81 = 4.11 / 9.81 = 0.42 g
Step 4, Angular velocity:
ω = v / r = 11.11 / 30 = 0.37 rad/s (≈ 3.54 rpm)
Why speed matters more than mass for cornering
In the formula F = mv²/r, velocity appears squared. Doubling your speed through a roundabout quadruples the required centripetal force, which is why speed limits in corners are so important. At 80 km/h on the same 30 m roundabout, the required force would be 19,753 N (4× higher), far exceeding normal tyre friction limits and causing a skid.
Understanding Centripetal Force
What Is Centripetal Force, and What Provides It?
Any object moving in a circle is constantly accelerating, not because its speed is changing, but because its direction of motion is changing. Newton's second law requires a net force to produce this acceleration, and that force must point toward the centre of the circle. This inward force is what physicists call centripetal force.
The key insight is that centripetal force is not a fundamental force of nature, it is a role that any real force can play. Different situations use different forces to fill that role:
- ›Gravity: keeps the Moon in orbit around Earth, and Earth in orbit around the Sun. The entire orbital dynamics of the solar system runs on gravity-as-centripetal-force.
- ›Tension: a ball on a string, a hammer throw, or a tether on a spinning space station all use cable tension directed inward.
- ›Friction: when you drive around a curve, the tyres grip the road and friction acts sideways toward the centre of the turn. This is the only centripetal force available on flat roads, which is why icy corners are dangerous.
- ›Normal force: on banked roads, racing tracks, and velodrome cycling tracks, the slope of the surface provides an inward component of the normal force, supplementing friction.
- ›Electromagnetic force: in particle accelerators, powerful magnetic fields bend charged particles into circular paths, centripetal force at the atomic scale.
Centripetal vs Centrifugal, The Common Misconception
The term "centrifugal force" is often heard but frequently misunderstood. Here is the critical distinction:
- ›Centripetal force is real: it is the actual inward force (gravity, tension, friction) that constrains the object to circular motion. It exists in any reference frame.
- ›Centrifugal force is a pseudo-force: it appears only in a rotating (non-inertial) reference frame. A passenger in a car feels "pushed outward" into the door, but in reality, the door is being pushed inward into the passenger. There is no outward force; the passenger's inertia wants to continue in a straight line.
- ›In an inertial (non-rotating) frame, the only frame where Newton's laws apply directly, there is only the centripetal force directed inward, and no centrifugal force at all.
- ›Engineers sometimes work in rotating frames (e.g. designing centrifuges, rotating space habitats) where the centrifugal pseudo-force is a useful mathematical tool, but it must always be labeled a fictitious force.
The classic demonstration
Swing a bucket of water in a vertical circle at sufficient speed. At the top of the loop, water stays in the bucket even though the bucket is upside-down. There is no outward force pushing the water in, instead, gravity and the bucket floor together provide the inward centripetal force, and the water's inertia keeps it "in place" relative to the bucket. Slow down below the critical speed (v = √(rg)) and gravity exceeds the required centripetal force, the water falls out.
Real-World Centripetal Force Applications
| Scenario | How centripetal force arises | Key value |
|---|---|---|
| Car on a banked curve | Road normal force provides centripetal acceleration toward curve centre | F = mv²/r → safe cornering speed |
| Roller coaster loop | Track normal force + gravity combine to keep rider in circular path | Minimum speed at top: v = √(rg) |
| Washing machine spin cycle | Drum wall pushes wet clothes inward; water escapes through holes outward | a_c = ω²r → effective g up to 300–1,000 g |
| Satellite orbit | Gravity provides all centripetal force; no engine thrust needed | F = GMm/r² = mv²/r → orbital velocity |
| Centrifuge in medicine | Separates blood components by density using centripetal acceleration | Typical: 1,000–15,000 × g |
| Aircraft banking | Lift component toward turn centre acts as centripetal force | F_c = L sin θ, where θ = bank angle |
| Cycling on a velodrome | Banked track and friction keep rider on circular path at high speed | Optimal angle = arctan(v²/rg) |
| ISS orbit | Earth gravity provides centripetal force at 408 km altitude | v ≈ 7.66 km/s, T ≈ 92.7 min |
G-Force, Centripetal Acceleration in Human Terms
G-force (or g-force) is centripetal acceleration expressed as a multiple of Earth's gravitational acceleration (g = 9.81 m/s²). It gives an intuitive sense of how strongly circular motion is felt as a body force:
g-force = a_c / 9.81 = (v² / r) / 9.81 = ω² × r / 9.81
| Situation | G-force | Notes |
|---|---|---|
| Standing still | 1 g | Normal resting weight; baseline |
| Fast elevator start | 1.2–1.5 g | Mild heaviness sensation |
| Sports car cornering | 1.0–1.5 g | Side-to-side force on driver |
| Fighter jet manoeuvre | 5–9 g | G-suit required; unconsciousness risk above 5–7 g |
| Roller coaster loop | 3–6 g | Rider feels heaviest at loop bottom |
| Rocket launch | 3–4 g | Sustained; astronauts trained for this level |
| Car crash at 48 km/h | 20–40 g | Brief impulse; crumple zones reduce duration |
| Medical centrifuge | 500–15,000 g | Separates blood cells from plasma |
| Washing machine spin | 300–1,000 g | Squeezes water from clothes by apparent weight |
The human body tolerates sustained g-forces up to about 5 g in the head-to-feet direction before blood pools in the legs and the brain loses oxygen (called G-LOC, G-force induced Loss Of Consciousness). Fighter pilots wear pressure suits and perform the anti-G straining manoeuvre (tensing leg and abdominal muscles) to raise this limit to 7–9 g.
Designing for Centripetal Force, Engineering Principles
- ›Bank angle formula: the ideal bank angle θ for a road curve at speed v and radius r is θ = arctan(v²/rg). At this angle, no friction is needed, the normal force alone provides centripetal force. Highway on-ramps are banked at 5–8° for typical speeds.
- ›Minimum loop speed: in a vertical loop of radius r, the minimum speed at the top to maintain contact is v_min = √(rg). Below this, gravity exceeds the required centripetal force and the rider leaves the track.
- ›Satellite orbital speed: combining F = mv²/r with F = GMm/r² gives v_orbital = √(GM/r). For Earth at ISS altitude (408 km): v ≈ 7.66 km/s.
- ›Centrifuge radius selection: for a target g-force at a fixed RPM, r = g × 9.81 / ω². Medical centrifuges trade radius for speed, shorter tubes spin faster to achieve the same separation force.
- ›Tyre friction limit: the maximum speed for a flat curve is v_max = √(μ × r × g), where μ is the coefficient of friction (≈ 0.7–1.0 for rubber on dry asphalt). This gives the speed at which the required centripetal force equals the maximum available friction force.
Frequently Asked Questions
What is centripetal force?
- ›Centripetal force is the net inward force that keeps an object moving in a circular path.
- ›It always points toward the centre of the circle, perpendicular to the velocity.
- ›It is not a new type of force, it is whatever real force (gravity, tension, friction, normal force) is directed inward.
- ›Without centripetal force, the object would travel in a straight line (Newton's first law).
- ›Formula: F = mv²/r, where m is mass, v is speed, and r is the radius of the circular path.
What is the formula for centripetal force?
- ›F = mv²/r, using linear velocity v (m/s), mass m (kg), and radius r (m)
- ›F = mω²r, using angular velocity ω (rad/s)
- ›F = m(2πf)²r, using frequency f (Hz)
- ›F = m(2π/T)²r, using period T (seconds)
- ›All four forms are mathematically equivalent; choose based on which velocity measure you know.
- ›The result is in Newtons (N) when SI units are used.
What is the difference between centripetal and centrifugal force?
- ›Centripetal force is real, it is the actual inward force (gravity, tension, friction) that causes circular motion.
- ›Centrifugal force is a pseudo-force, it appears only in a rotating reference frame and does not exist in an inertial (non-rotating) frame.
- ›When you feel "pushed outward" in a turning car, that is your inertia resisting the change in direction, not an outward force.
- ›The door pushes you inward (centripetal); your body interprets this as being pushed outward (centrifugal sensation).
- ›Engineers sometimes use centrifugal force as a mathematical tool in rotating reference frames, but it is always labeled fictitious.
How does centripetal force apply to satellite orbits?
- ›For a satellite in circular orbit, gravity provides all the centripetal force.
- ›Setting F_gravity = F_centripetal: GMm/r² = mv²/r → v = √(GM/r)
- ›The orbital speed depends only on the central mass (M) and orbital radius (r), not the satellite mass.
- ›Higher orbits require slower speeds: the ISS at 408 km orbits at 7.66 km/s; geostationary satellites at 35,786 km orbit at 3.07 km/s.
- ›If gravity fell away, the satellite would travel in a straight tangential line, this is Newton's first law in action.
Why does doubling the speed require four times the centripetal force?
- ›In the formula F = mv²/r, velocity is squared.
- ›If v doubles (2v), then v² becomes (2v)² = 4v², so F quadruples.
- ›This is why speed limits on curves are so critical, a small speed increase dramatically raises the required centripetal force.
- ›Example: cornering at 40 km/h needs 4,937 N; at 80 km/h the same turn needs 19,748 N (4×).
- ›This also explains why race cars, which corner at high speed, need wide tyres, banked tracks, and aerodynamic downforce to maintain grip.
What is centripetal acceleration, and how is it related to g-force?
- ›Centripetal acceleration a_c = v²/r = ω²r = F/m (in m/s²)
- ›It is always directed toward the centre of the circle.
- ›G-force = a_c / 9.81, a dimensionless multiple of Earth's surface gravity.
- ›1 g is what you feel standing still on Earth's surface.
- ›Sports car cornering: ~1.0–1.5 g; roller coaster loop bottom: ~3–6 g; fighter jet manoeuvre: up to 9 g.
- ›G-LOC (G-force induced Loss Of Consciousness) typically occurs above 5–7 g in the head-to-feet direction.
How does a washing machine use centripetal force to remove water?
- ›The rotating drum forces wet clothes to travel in a circle, the drum wall pushes inward (centripetal force).
- ›Water molecules are less constrained, when they reach the perforations in the drum, they have no inward force and fly off tangentially (not outward, tangentially).
- ›The drum's centripetal acceleration is ω²r. A typical spin of 1,200 rpm at r = 0.25 m gives a_c = (1200 × π/30)² × 0.25 ≈ 1,974 m/s² ≈ 201 g.
- ›At 201 g, water experiences an apparent weight 201 times greater than normal, squeezing it out far more efficiently than gravity alone.
- ›Higher spin speeds (1,400–1,600 rpm) or larger drum radii increase the g-force further, removing more water per cycle.
What is the minimum speed to complete a vertical loop without falling?
- ›At the top of a vertical loop, gravity acts downward (toward centre) and provides centripetal force.
- ›The minimum speed condition is when the normal force from the track equals zero: mg = mv²/r → v_min = √(rg).
- ›Below this speed, gravity exceeds the required centripetal force and the rider leaves the track.
- ›Example: for a loop of radius 5 m, v_min = √(5 × 9.81) = 7.0 m/s (≈ 25.2 km/h).
- ›Roller coasters always exceed this minimum speed with a safety margin; loop entry speed is typically 1.5–2× v_min.
- ›G-force at the top of the loop at minimum speed = 1 g; at higher speeds, it increases above 1 g.
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