F = ma calculator: force, friction and inclined plane

How to calculate force with F = ma

Newton's second law says net force equals mass times acceleration: F = ma, so a = F/m. A 3,000 N net force on a 1,500 kg car gives 3,000 ÷ 1,500 = 2 m/s², about 0.2 g. With friction, subtract it from the push first: on level ground, a = (F − μk·mg)/m.

What counts as the net force

Opposing forces subtract. If a car's tires push forward with 3,500 N while drag and rolling resistance push back with 500 N, the net force is 3,000 N and the car accelerates at 2 m/s².

Weight is a separate force. The 1,500 kg car weighs 1,500 × 9.80665 = 14,710 N, but on level ground the road's normal force cancels it, so it leaves the horizontal acceleration unchanged.

One newton gives 1 kg an acceleration of 1 m/s². A pound-force is 4.4482216152605 N, the weight of 1 lb under standard gravity, so 100 lbf accelerating an object at 1 g means a mass of 100 lb, or 45.36 kg.

How to calculate friction force

Friction is the coefficient of friction times the normal force: f = μN. On level ground N = mg, so sliding a 25 kg wooden box across a wooden floor with μk = 0.3 takes 0.3 × 25 × 9.80665 = 73.55 N at constant speed.

Push that box with 100 N and the net force is 100 − 73.55 = 26.45 N, so it accelerates at 26.45 ÷ 25 = 1.06 m/s².

Static friction is a limit, not a fixed value. It matches whatever force tries to start the motion, up to μs × N. Beyond that the object slips, and kinetic friction, μk × N, takes over.

Approximate coefficients from OpenStax University Physics, Table 6.1, 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

Surface finish, moisture and temperature all move these numbers, so a coefficient measured on your own surfaces is more reliable.

Block on an inclined plane with friction

On a slope at angle θ, split the weight mg into mg sin θ pulling down the slope and mg cos θ pressing into it. The normal force balances the second part, N = mg cos θ, so kinetic friction is μk mg cos θ.

For a 10 kg block on a 30° slope with μk = 0.2:

1. Weight: W = 10 × 9.80665 = 98.07 N.
2. Down-slope pull: 98.07 × sin 30° = 49.03 N.
3. Normal force: 98.07 × cos 30° = 84.93 N.
4. Kinetic friction: 0.2 × 84.93 = 16.99 N.
5. Net force: 49.03 − 16.99 = 32.05 N, so a = 32.05 ÷ 10 = 3.20 m/s² down the slope.

In one line, a = g(sin θ − μk cos θ). Mass cancels, so a heavier block slides no faster when friction is proportional to its weight.

Will the block slide? The angle of repose

A block at rest starts to slide when the down-slope pull beats the most static friction can supply: mg sin θ > μs mg cos θ, which simplifies to tan θ > μs. The critical angle is θc = arctan μs.

For wood on wood, μs = 0.5 gives θc = 26.57°. A 10 kg wooden block holds at 20°, where friction supplies just the 33.54 N needed. At 30° it slides, accelerating at 2.36 m/s² with μk = 0.3.

While the block holds, friction equals the down-slope pull, not μs N. On a 30° slope with μs = 0.7, the 10 kg block needs 49.03 N of friction, under the 59.45 N limit (HyperPhysics).

Acceleration and net force compared

Default is 3,000 N on 1,500 kg. Slope is the 10 kg, 30°, μk = 0.2 block worked above. Static holds is the same block with μk = 0.5 and μs = 0.7, where friction cancels the pull and the net force is zero.

Common mistakes

• Using mg as the normal force on a slope. N is mg cos θ. Using 98.07 N instead of 84.93 N in the example gives 2.94 m/s² instead of 3.20 m/s².
• Setting static friction to μs N. A resting block only gets the friction it needs.
• Confusing pounds with pound-force. A pound is mass; a pound-force is 4.448 N of force.
• Using μk to decide whether a block starts to move. Starting depends on μs, which is higher for most surface pairs.
• Measuring the slope from vertical. A 30° angle from vertical is a 60° slope, where the same block accelerates at 7.51 m/s².

Units, signs and coefficients

• Force can be in N, kN, lbf or kgf, and acceleration in m/s², ft/s² or multiples of standard gravity.
• Force and acceleration can be negative to show direction; mass must be positive.
• The static coefficient is optional. Left blank, μk sets the angle at which the block starts to slide.
• Results can be shown in SI (N, kg, m/s²) or US customary units (lbf, lb, ft/s²).

To turn a constant acceleration into distance and time, use the kinematics calculator. For the inward force on a curved path, use the circular motion calculator.