# 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.

परस्परसंवादी आवृत्ती: https://www.calcopenly.com/mr/science/circular-motion-rotation-calculator
विषय: विज्ञानाचे कॅल्क्युलेटर

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.

## इनपुट

- **Calculate** (पर्याय: Centripetal force, Angular velocity, Moment of inertia, बलाघूर्ण)
- **Solve for** (पर्याय: बल, वेग, त्रिज्या)
- **वस्तुमान**
- **वेग**
- **Radius of the circle**
- **Centripetal force**
- **Rotation rate**
- **Mass of the body**
- **Shape and axis** (पर्याय: 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)**

## निकाल

- Centripetal force (N) — मुख्य निकाल
- Centripetal acceleration (m/s²)
- वेग (m/s)
- वेग (km/h)
- त्रिज्या (m)
- Angular velocity (rad/s)
- Revolutions per minute (rpm)
- वारंवारता (Hz)
- Period (one revolution) (s)
- Moment of inertia (kg·m²)
- Rotational kinetic energy (J)
- Angular momentum (kg·m²/s)
- बलाघूर्ण (N·m)
- Angular acceleration (rad/s²)
- Centripetal acceleration (g)

## सूत्र

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

## सोडवलेली उदाहरणे

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

- Calculate: Centripetal force
- Solve for: बल
- वस्तुमान: 1000 kg
- वेग: 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**
- पडताळणीचा स्रोत: 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: वेग
- वस्तुमान: 10 kg
- Radius of the circle: 2 m
- Centripetal force: 500 N
- **वेग: 10 m/s**
- पडताळणीचा स्रोत: 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**
- **वारंवारता: 50 Hz**
- **Period (one revolution): 0.02 s**
- **वेग: 31.4159 m/s**
- पडताळणीचा स्रोत: 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**
- पडताळणीचा स्रोत: 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**
- पडताळणीचा स्रोत: 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: बलाघूर्ण
- 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
- **बलाघूर्ण: 15 N·m**
- **Angular acceleration: 30 rad/s²**
- पडताळणीचा स्रोत: Python 3.8: τ = 0.3 × 50 × sin90° = 15; α = τ/I = 30

## प्रश्न

### 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.

### “Centripetal force, moment of inertia and torque calculator” किती अचूक आहे?

अचूकता तुमच्या इनपुटवर आणि पद्धतीच्या गृहीतकांवर अवलंबून असते. दशांश गणना 50 सार्थ अंक वापरते, पण अंदाज, संख्यात्मक पद्धती आणि मूळ डेटा कमी अचूक असू शकतात; दाखवलेल्या मूल्यांचे पूर्णांकन केल्याने या मर्यादा दूर होत नाहीत. स्वतंत्र स्रोतांतील सोडवलेल्या उदाहरणांशी पडताळणी: ७. उदाहरणार्थ, “1000 kg car at 20 m/s on a 50 m curve” ची पडताळणी Python 3.8 decimal: a = 400/50 = 8, F = 8000, ω = v/r = 0.4 याच्याशी केली आहे.

### या पद्धतीचा स्रोत कोणता?

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.

## स्रोत

- [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