# 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.

النسخة التفاعلية: https://www.calcopenly.com/ar/science/circular-motion-rotation-calculator
الموضوع: حاسبات العلوم

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.

## المدخلات

- **Calculate** (الخيارات: Centripetal force, Angular velocity, Moment of inertia, عزم الدوران)
- **Solve for** (الخيارات: القوة, السرعة, نصف القطر)
- **الكتلة**
- **السرعة**
- **Radius of the circle**
- **Centripetal force**
- **Rotation rate**
- **Mass of the body**
- **Shape and axis** (الخيارات: 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)**

## النتائج

- Centripetal force (N) — النتيجة الرئيسية
- Centripetal acceleration (m/s²)
- السرعة (m/s)
- السرعة (km/h)
- نصف القطر (m)
- Angular velocity (rad/s)
- Revolutions per minute (rpm)
- التردد (Hz)
- Period (one revolution) (s)
- Moment of inertia (kg·m²)
- Rotational kinetic energy (J)
- Angular momentum (kg·m²/s)
- عزم الدوران (N·m)
- Angular acceleration (rad/s²)
- Centripetal acceleration (g)

## الصيغة

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

## أمثلة محلولة

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

- Calculate: Centripetal force
- Solve for: القوة
- الكتلة: 1000 kg
- السرعة: 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**
- مصدر التحقق: ⁨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: السرعة
- الكتلة: 10 kg
- Radius of the circle: 2 m
- Centripetal force: 500 N
- **السرعة: 10 m/s**
- مصدر التحقق: ⁨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**
- **التردد: 50 Hz**
- **Period (one revolution): 0.02 s**
- **السرعة: 31.4159 m/s**
- مصدر التحقق: ⁨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**
- مصدر التحقق: ⁨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**
- مصدر التحقق: ⁨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: عزم الدوران
- 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
- **عزم الدوران: 15 N·m**
- **Angular acceleration: 30 rad/s²**
- مصدر التحقق: ⁨Python 3.8: τ = 0.3 × 50 × sin90° = 15; α = τ/I = 30⁩

## الأسئلة

### 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.

### ما مدى دقة «⁨Centripetal force, moment of inertia and torque calculator⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 7. مثلًا، يجري التحقق من «⁨1000 kg car at 20 m/s on a 50 m curve⁩» بالرجوع إلى ⁨Python 3.8 decimal: a = 400/50 = 8, F = 8000, ω = v/r = 0.4⁩.

### ما مصدر هذه الطريقة؟

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.

## المصادر

- [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