# 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.

Phiên bản tương tác: https://www.calcopenly.com/vi/science/circular-motion-rotation-calculator
Chủ đề: Máy tính khoa học theo chủ đề

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.

## Dữ liệu đầu vào

- **Calculate** (lựa chọn: Centripetal force, Angular velocity, Moment of inertia, Mômen xoắn)
- **Solve for** (lựa chọn: Lực, Tốc độ, Bán kính)
- **Khối lượng**
- **Tốc độ**
- **Radius of the circle**
- **Centripetal force**
- **Rotation rate**
- **Mass of the body**
- **Shape and axis** (lựa chọn: 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)**

## Kết quả

- Centripetal force (N) — kết quả chính
- Centripetal acceleration (m/s²)
- Tốc độ (m/s)
- Tốc độ (km/h)
- Bán kính (m)
- Angular velocity (rad/s)
- Revolutions per minute (rpm)
- Tần số (Hz)
- Period (one revolution) (s)
- Moment of inertia (kg·m²)
- Rotational kinetic energy (J)
- Angular momentum (kg·m²/s)
- Mômen xoắn (N·m)
- Angular acceleration (rad/s²)
- Centripetal acceleration (g)

## Công thức

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

## Ví dụ có lời giải

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

- Calculate: Centripetal force
- Solve for: Lực
- Khối lượng: 1000 kg
- Tốc độ: 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**
- Nguồn đối chiếu: 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: Tốc độ
- Khối lượng: 10 kg
- Radius of the circle: 2 m
- Centripetal force: 500 N
- **Tốc độ: 10 m/s**
- Nguồn đối chiếu: 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**
- **Tần số: 50 Hz**
- **Period (one revolution): 0.02 s**
- **Tốc độ: 31.4159 m/s**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: 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: Mômen xoắn
- 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
- **Mômen xoắn: 15 N·m**
- **Angular acceleration: 30 rad/s²**
- Nguồn đối chiếu: Python 3.8: τ = 0.3 × 50 × sin90° = 15; α = τ/I = 30

## Câu hỏi

### 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” chính xác đến mức nào?

Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 7. Ví dụ, “1000 kg car at 20 m/s on a 50 m curve” được kiểm tra bằng Python 3.8 decimal: a = 400/50 = 8, F = 8000, ω = v/r = 0.4.

### Phương pháp này lấy từ đâu?

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.

## Nguồn

- [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