Physics
Dynamics and Newton's laws calculator
Newton's second law, weight, friction, the inclined plane, centripetal force and impulse. Pick the relation, leave blank the quantity the exercise asks for, and the formula inverts itself.
The three laws, and where they are actually used
The first law says that with no resultant force the motion does not change: a body at rest stays at rest, one in motion carries on in a straight line at constant speed. It is not a special case of the second — it is the definition of which frames of reference are entitled to use it. The second, F = ma, is the one you calculate with: the resultant of all the forces, not any one of them, is what stands on the left. The third says forces come in equal and opposite pairs applied to different bodies — and it is the word different that stops them cancelling out.
The inclined plane is where the second law is seen most clearly, because the weight has to be resolved: a component along the slope, mg·sin θ, which pulls the body down it, and one perpendicular, mg·cos θ, which presses and produces the normal reaction. Friction is μ times that reaction, and putting it all into F = ma the mass cancels: a = g(sin θ − μ cos θ). If the angle is less than arctan μ the answer is negative, which does not mean the body climbs but that it never starts.
Centripetal force is the one that causes most confusion, because it is not an extra force to add to the diagram. A car in a bend is held on its line by friction; a stone in a sling by the tension in the cord; the Moon by gravity. “Centripetal” names the job, not the cause, and it is the cause that goes in the force diagram.
Common mistakes
- Putting a single force into F = ma instead of the resultant: if a body is pulled and braked at once, the difference is what goes on the left.
- Confusing mass and weight: weight is in newtons and changes with gravity, mass is in kilograms and does not.
- Using mg for the normal reaction on a slope: there it is mg·cos θ, always smaller.
- Adding the centripetal force to the others in the diagram: it is already one of them, and counting it twice doubles the resultant.
Frequently asked questions
Why does the mass disappear on an inclined plane?
Because it appears on both sides of the equation. The component of the weight along the slope is mg·sin θ and friction is μmg·cos θ: put them into ma and divide through by m, and the mass is gone. It is the same reason a feather and a hammer fall together in a vacuum.
What is the difference between static and kinetic friction?
Static friction is what stops the body from starting, and is at most μₛN: as long as the applied force stays below that threshold, friction balances it exactly. Kinetic friction acts once the body is already sliding, is μₖN and is usually a little smaller. This page uses a single coefficient, which is the textbook approximation.
Is centripetal force a real force?
Yes, but it is not a force of its own: it is the name given to the resultant pointing at the centre, whatever produces it. Not to be confused with centrifugal force, which is apparent and appears only if you choose a rotating frame.
What does impulse measure?
How much a force changes momentum. The same impulse comes from a large force for a short time or a small one for long: which is why an airbag, by stretching the impact out, reduces the force for the same change of speed.
Which value of g should I use?
9.81 m/s² is the average at the Earth's surface and is right wherever an exercise does not say otherwise. Some textbooks use 9.8 or even 10 to keep the arithmetic simple; the field is editable for exactly that reason.
How this calculation works
Second law: F = m·a, inverted to m = F/a and a = F/m. Weight: W = m·g. Sliding friction: Fᶠ = μ·N. Inclined plane: resolving the weight into mg·sin θ along the slope and mg·cos θ perpendicular to it, and with friction equal to μ times the second, the second law gives a = g(sin θ − μ cos θ), where the mass has cancelled. Recovering the angle from an acceleration has no closed form — θ sits inside both a sine and a cosine — so it is found by bisection over the quarter turn where the acceleration rises with the angle. Centripetal force: F = m·v²/r, giving the acceleration v²/r, the angular speed v/r and the period 2πr/v. Impulse: J = F·Δt, numerically the change of momentum.
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