# Centripetal force, moment of inertia and torque calculator

> Centripetal force and acceleration from mass, speed and radius; rpm to rad/s and period; moment of inertia of nine shapes; torque from force and lever arm.

Version interactive : https://www.calcopenly.com/fr/science/circular-motion-rotation-calculator
Sujet : Calculatrices scientifiques

Centripetal force is the inward pull that keeps a mass on a circular path: F = mv²/r, with the centripetal acceleration a = v²/r = ω²r pointing at the centre. The other modes convert a rotation rate between rpm, rad/s, hertz and period (ω = 2πf), give the moment of inertia I of nine standard shapes, and work out torque τ = rF sin θ and angular acceleration α = τ/I.

The default is a 1,000 kg car taking a 50 m radius curve at 20 m/s (72 km/h). The tyres must supply 8,000 N of sideways grip, an acceleration of 8 m/s² or 0.816 g. In torque mode, a 50 N push at 90° on a 0.3 m wrench gives 15 N·m.

Results assume uniform circular motion and rigid bodies of uniform density. Mass farther from the axis raises I, so a thin hoop has twice the moment of inertia of a solid disc with the same mass and radius.

## Données

- **Calculate** (options : Centripetal force, Angular velocity, Moment of inertia, Couple)
- **Solve for** (options : Force, Vitesse, Rayon)
- **Masse**
- **Vitesse**
- **Radius of the circle**
- **Centripetal force**
- **Rotation rate**
- **Mass of the body**
- **Shape and axis** (options : Point mass at radius R, Thin hoop or thin-walled cylinder (central axis), Solid cylinder or disc (central axis), Thick-walled cylinder (central axis), Solid sphere (through centre), Thin spherical shell (through centre), Thin rod about its centre, Thin rod about one end, Rectangular plate (axis through centre, perpendicular))
- **Radius R (outer R₁ for thick cylinder)**
- **Inner radius R₂**
- **Length L (side a for plate)**
- **Side b**
- **Spin rate (for rotational energy)**
- **Lever arm r (axis to point of force)**
- **Applied force**
- **Angle between arm and force**
- **Moment of inertia (for angular acceleration)**

## Résultats

- Centripetal force (N) — résultat principal
- Centripetal acceleration (m/s²)
- Vitesse (m/s)
- Vitesse (km/h)
- Rayon (m)
- Angular velocity (rad/s)
- Revolutions per minute (rpm)
- Fréquence (Hz)
- Period (one revolution) (s)
- Moment of inertia (kg·m²)
- Rotational kinetic energy (J)
- Angular momentum (kg·m²/s)
- Couple (N·m)
- Angular acceleration (rad/s²)
- Centripetal acceleration (g)

## Formule

$$
a_c = \frac{v^2}{r} = \omega^2 r,\quad F_c = \frac{mv^2}{r},\quad \omega = 2\pi f = \frac{2\pi\,\text{rpm}}{60},\quad I = \sum m r^2,\quad \tau = rF\sin\theta = I\alpha
$$

## Exemples détaillés

### 1000 kg car at 20 m/s on a 50 m curve

- Calculate: Centripetal force
- Solve for: Force
- Masse: 1000 kg
- Vitesse: 20 m/s
- Radius of the circle: 50 m
- **Centripetal force: 8,000 N**
- **Centripetal acceleration: 8 m/s²**
- **Angular velocity: 0.4 rad/s**
- Source de vérification : Python 3.8 decimal: a = 400/50 = 8, F = 8000, ω = v/r = 0.4

### Speed that needs 500 N on 10 kg at 2 m

- Calculate: Centripetal force
- Solve for: Vitesse
- Masse: 10 kg
- Radius of the circle: 2 m
- Centripetal force: 500 N
- **Vitesse: 10 m/s**
- Source de vérification : Python 3.8 decimal: v = √(Fr/m) = √100

### 3000 rpm at 10 cm

- Calculate: Angular velocity
- Radius of the circle: 0.1 m
- Rotation rate: 3000 rpm
- **Angular velocity: 314.159 rad/s**
- **Fréquence: 50 Hz**
- **Period (one revolution): 0.02 s**
- **Vitesse: 31.4159 m/s**
- Source de vérification : Python 3.8 math: ω = 2π·3000/60, v = ωr

### Solid sphere 5 kg, R = 0.2 m

- Calculate: Moment of inertia
- Mass of the body: 5 kg
- Shape and axis: Solid sphere (through centre)
- Radius R (outer R₁ for thick cylinder): 0.2 m
- **Moment of inertia: 0.08 kg·m²**
- Source de vérification : Serway Table 10.2: I = 2/5 MR² = 0.4 × 5 × 0.04

### Rod about its end, 2 kg × 3 m, spinning at 60 rpm

- Calculate: Moment of inertia
- Mass of the body: 2 kg
- Shape and axis: Thin rod about one end
- Length L (side a for plate): 3 m
- Spin rate (for rotational energy): 60 rpm
- **Moment of inertia: 6 kg·m²**
- **Rotational kinetic energy: 118.435 J**
- **Angular momentum: 37.6991 kg·m²/s**
- Source de vérification : Python 3.8 math: I = ML²/3 = 6; ω = 2π; KE = ½Iω² = 12π² = 118.4353; L = 12π

### Wrench: 50 N at 90° on a 0.3 m arm

- Calculate: Couple
- Lever arm r (axis to point of force): 0.3 m
- Applied force: 50 N
- Angle between arm and force: 90 °
- Moment of inertia (for angular acceleration): 0.5 kg·m²
- **Couple: 15 N·m**
- **Angular acceleration: 30 rad/s²**
- Source de vérification : Python 3.8: τ = 0.3 × 50 × sin90° = 15; α = τ/I = 30

## Questions

### What is the formula for centripetal force?

F = mv²/r, with mass m in kilograms, speed v in metres per second and radius r in metres, giving newtons. Equivalent forms are F = mω²r and F = 4π²mr/T². Doubling the speed quadruples the force and doubling the radius halves it: a 1,000 kg car at 20 m/s on a 50 m curve needs 8,000 N, and at 40 m/s it would need 32,000 N.

### How do you convert rpm to rad/s?

Multiply rpm by 2π/60, about 0.10472. 3,000 rpm is 314.159 rad/s, or 50 revolutions per second (50 Hz), so one revolution takes 0.02 s. To go back, multiply rad/s by 60/(2π), about 9.5493. The radian is the coherent SI unit of plane angle (BIPM SI Brochure), so rad/s is the SI unit of angular velocity.

### Is centrifugal force real?

Not in an inertial, non-rotating frame of reference: there, the only horizontal force on a cornering car is the inward centripetal force from the tyres. Centrifugal force appears only when motion is described from inside the rotating frame, where it has the same size, mv²/r, pointing outward. The outward push a passenger feels is their body's inertia carrying it in a straight line while the car turns.

### What is the moment of inertia of a solid disc?

I = ½MR² about its central axis, so a 5 kg disc of radius 0.2 m has I = 0.5 × 5 × 0.2² = 0.1 kg·m². Other standard results are MR² for a thin hoop, ⅖MR² for a solid sphere, ⅔MR² for a thin spherical shell, and ML²/12 or ML²/3 for a thin rod about its centre or one end (Serway and Jewett, Table 10.2).

### How do you calculate torque?

Torque is τ = rF sin θ, where r is the distance from the axis to the point where the force acts and θ is the angle between the arm and the force. A 50 N push at 90° on a 0.3 m wrench gives 15 N·m; the same push along the handle (θ = 0°) gives none. One newton-metre is 0.7376 pound-force feet.

### Quelle est la précision de « Centripetal force, moment of inertia and torque calculator » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 7. Par exemple, « 1000 kg car at 20 m/s on a 50 m curve » est vérifié à l’aide de Python 3.8 decimal: a = 400/50 = 8, F = 8000, ω = v/r = 0.4.

### D’où vient cette méthode ?

OpenStax University Physics Volume 1, §6.3 Centripetal force; §10.5 Calculating moments of inertia; §10.6 Torque; Serway & Jewett, Physics for Scientists and Engineers, Table 10.2 — Moments of inertia of homogeneous rigid objects.

## Sources

- [OpenStax University Physics Volume 1, §6.3 Centripetal force; §10.5 Calculating moments of inertia; §10.6 Torque](https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia)
- Serway & Jewett, Physics for Scientists and Engineers, Table 10.2 — Moments of inertia of homogeneous rigid objects