# F = ma calculator: force, friction and inclined plane

> Solve Newton's second law F = ma for force, mass or acceleration, or find acceleration, normal force and friction for a block on an inclined plane.

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主题：科学计算器合集

Newton's second law says the net force on a body equals its mass times its acceleration, F = ma, so one newton gives 1 kg an acceleration of 1 m/s². Enter two of force, mass and acceleration to get the third. The inclined-plane mode splits the block's weight into mg sin θ down the slope and mg cos θ into it; the normal force balances the second part, and kinetic friction μk × N opposes the sliding.

The default, 3,000 N on a 1,500 kg car, gives 2 m/s², about 0.2 g. On the slope, a 10 kg block on a 30° incline with μk = 0.2 slides down at 3.20 m/s² against 16.99 N of friction.

Friction coefficients depend on both surfaces and their condition, so treat slope results as estimates. The block stays put while tan θ is at or below the static coefficient μs; leave μs blank and μk is used as that threshold.

## 输入

- **Problem** (选项：F = ma, Inclined plane)
- **Solve for** (选项：力, 质量, 加速度)
- **Net force F**
- **Mass m**
- **Acceleration a**
- **Block mass**
- **Incline angle**
- **Kinetic friction coefficient μk**: Approximate kinetic values from OpenStax University Physics Table 6.1
- **Static friction coefficient μs**: Leave blank to use μk as the threshold for starting to slide
- **Gravitational acceleration**
- **结果单位** (选项：SI, US customary)

## 结果

- 加速度 (m/s²) — 主要结果
- Net force (N)
- 质量 (kg)
- Block
- Normal force (N)
- Friction force (N)
- Weight component down the slope (N)
- Weight (N)
- Angle at which it starts to slide (°)

## 公式

$$
F = ma;\qquad N = mg\cos\theta,\quad a = g(\sin\theta - \mu_k\cos\theta)\ \text{when } \tan\theta > \mu_s
$$

## 计算示例

### 3000 N on a 1500 kg car

- Problem: F = ma
- Solve for: 加速度
- Net force F: 3000 N
- Mass m: 1500 kg
- 结果单位: SI
- **加速度: 2 m/s²**
- 核验来源：Python 3.8: a = F/m = 3000/1500

### Force to give 2 kg an acceleration of 1 g

- Problem: F = ma
- Solve for: 力
- Mass m: 2 kg
- Acceleration a: 1 g
- 结果单位: SI
- **Net force: 19.6133 N**
- 核验来源：Python 3.8: 2 × 9.80665 = 19.6133

### 100 lbf gives 1 g: mass is 100 lb (45.359237 kg)

- Problem: F = ma
- Solve for: 质量
- Net force F: 100 lbf
- Acceleration a: 1 g
- 结果单位: SI
- **质量: 45.3592 kg**
- 核验来源：Definition of the pound-force (NIST SP 811 B.8): 1 lbf = 0.45359237 kg × 9.80665 m/s²

### 10 kg block on a 30° slope, μk = 0.2

- Problem: Inclined plane
- Block mass: 10 kg
- Incline angle: 30 °
- Kinetic friction coefficient μk: 0.2
- 结果单位: SI
- **加速度: 3.20476 m/s²**
- **Normal force: 84.9281 N**
- **Friction force: 16.9856 N**
- **Block: Slides down**
- 核验来源：Python 3.8 math: N = mg cos30° = 84.92808…, f = 0.2N, a = g(sin30° − 0.2 cos30°) = 3.2047634

### Static friction holds (μs = 0.7 > tan 30°)

- Problem: Inclined plane
- Block mass: 10 kg
- Incline angle: 30 °
- Kinetic friction coefficient μk: 0.5
- Static friction coefficient μs: 0.7
- 结果单位: SI
- **加速度: 0 m/s²**
- **Friction force: 49.0333 N**
- **Block: Stays at rest**
- **Angle at which it starts to slide: 34.99 °**
- 核验来源：Python 3.8 math: tan30° = 0.577 < 0.7, friction = mg sin30° = 49.03325 N; atan(0.7) = 34.992°

### Frictionless slope

- Problem: Inclined plane
- Block mass: 5 kg
- Incline angle: 45 °
- Kinetic friction coefficient μk: 0
- 结果单位: SI
- **加速度: 6.93435 m/s²**
- **Friction force: 0 N**
- 核验来源：Python 3.8 math: a = g sin45° = 6.9343487

## 常见问题

### What is one newton of force?

One newton is the force that gives a 1 kg mass an acceleration of 1 m/s², so 1 N = 1 kg·m/s² in the SI. A 1 kg mass weighs 9.80665 N under standard gravity, and one pound-force is exactly 4.4482216152605 N (NIST SP 811). A 100 lbf push that accelerates an object at 1 g therefore means a mass of 100 lb, or 45.36 kg.

### How do you calculate the friction force?

Friction equals the coefficient of friction times the normal force, f = μN. On level ground N = mg, so a 10 kg crate with μk = 0.2 needs a 19.61 N push to keep sliding at constant speed. Static friction is a limit, not a fixed value: it matches the applied force up to μs × N, and the object starts to move once that limit is passed.

### At what angle does an object start to slide down a slope?

It starts to slide once the slope angle exceeds arctan μs, the point where the pull down the slope, mg sin θ, overtakes the most static friction can supply, μs mg cos θ. With μs = 0.7 that angle is 34.99°; with μs = 0.2 it is 11.31°. Tilting a surface until an object slips and reading the angle is a standard way to measure μs.

### What are typical coefficients of friction?

OpenStax University Physics Table 6.1 gives approximate values, static then kinetic: rubber on dry concrete 1.0 and 0.7, wood on wood 0.5 and 0.3, ice on ice 0.1 and 0.03. Real values shift with surface finish, moisture and temperature, so a measured coefficient beats a table value. The kinetic coefficient is lower than the static one for each of these pairs.

### Does a heavier block slide down a slope faster?

No. Mass cancels in a = g(sin θ − μk cos θ), because both the pull down the slope and the friction are proportional to the weight. A 5 kg block and a 50 kg block on a frictionless 45° slope both accelerate at 6.93 m/s². Differences seen in practice come from air resistance and from friction coefficients that change with load or speed.

### “F = ma calculator: force, friction and inclined plane”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：6。 例如，“3000 N on a 1500 kg car”根据Python 3.8: a = F/m = 3000/1500进行核验。

### 这种方法出自哪里？

OpenStax University Physics Volume 1, §5.3 Newton's second law and §6.2 Friction; HyperPhysics — Inclined plane with friction.

## 来源

- [OpenStax University Physics Volume 1, §5.3 Newton's second law and §6.2 Friction](https://openstax.org/books/university-physics-volume-1/pages/6-2-friction)
- [HyperPhysics — Inclined plane with friction](http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/incline.html)