# 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.

操作できるページ：https://www.calcopenly.com/ja/science/force-friction-incline-calculator
分野：科学の計算機

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.

## 入力

- **Problem** (選択肢：F = ma, Inclined plane)
- **Solve for** (選択肢：力, 質量, 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**
- **結果の単位** (選択肢：SI, US customary)

## 結果

- 加速度 (m/s²) — 主な結果
- Net force (N)
- 質量 (kg)
- Block
- Normal force (N)
- Friction force (N)
- Weight component down the slope (N)
- Weight (N)
- Angle at which it starts to slide (°)

## 計算式

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

## 計算例

### 3000 N on a 1500 kg car

- Problem: F = ma
- Solve for: 加速度
- Net force F: 3000 N
- Mass m: 1500 kg
- 結果の単位: SI
- **加速度: 2 m/s²**
- 照合元：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
- 結果の単位: SI
- **Net force: 19.6133 N**
- 照合元：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: 質量
- Net force F: 100 lbf
- Acceleration a: 1 g
- 結果の単位: SI
- **質量: 45.3592 kg**
- 照合元：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
- 結果の単位: SI
- **加速度: 3.20476 m/s²**
- **Normal force: 84.9281 N**
- **Friction force: 16.9856 N**
- **Block: Slides down**
- 照合元：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
- 結果の単位: 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 °**
- 照合元：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
- 結果の単位: SI
- **加速度: 6.93435 m/s²**
- **Friction force: 0 N**
- 照合元：Python 3.8 math: a = g sin45° = 6.9343487

## よくある質問

### 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」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：6。 例えば、「3000 N on a 1500 kg car」はPython 3.8: a = F/m = 3000/1500と照合しています。

### この計算方法の出典は何ですか？

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

## 出典

- [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)