Forces and momentum

📚The physics

Free-body diagrams come first. Draw the body as a dot, then one arrow for every force acting on it, and nothing else. No velocity arrows, no acceleration arrows, no forces the body exerts on other things. If you cannot name what is exerting a force, that force does not exist. Most force questions become arithmetic once the diagram is right.

Newton’s first law says a body keeps constant velocity unless a resultant force acts. Constant velocity includes being at rest, and it includes moving at a steady speed in a straight line — but not moving in a circle at steady speed, because velocity is a vector and its direction is changing.

Newton’s second law is \( F = ma \) when the mass is constant, but the more general and more useful form is \( F = \dfrac{\Delta p}{\Delta t} \): the resultant force equals the rate of change of momentum. Use the second form whenever mass is changing or whenever the question is about impulse.

Newton’s third law is the one most often stated correctly and used wrongly. A third-law pair acts on two different bodies, is the same type of force, and is equal in magnitude and opposite in direction. The weight of a book on a table and the normal force from the table are not a third-law pair — they act on the same body, and one is gravitational while the other is electromagnetic. The partner of the book’s weight is the gravitational pull the book exerts on the Earth.

Friction. Static friction adjusts itself up to a maximum of \( \mu_s N \); it is whatever it needs to be to prevent sliding, which is why you cannot simply write \( f = \mu_s N \) for a stationary block. Once sliding begins, kinetic friction is \( \mu_k N \) and \( \mu_k < \mu_s \).

Circular motion is not a new force. Centripetal force is a role, not a source. Something real always plays it: tension for a ball on a string, gravity for a satellite, friction for a car cornering, the normal force for a wall-of-death rider. Never add a separate centripetal force to a free-body diagram — identify which of the forces already there is doing the job. The magnitude required is

There is no centrifugal force in an inertial frame. What passengers feel in a cornering car is their own inertia and the seat pushing them inwards.

Momentum and impulse. Momentum \( p = mv \) is a vector, so signs matter and you must choose a positive direction before you start. Impulse is the change in momentum, \( J = \Delta p = F\Delta t \) for a constant force, and for a varying force it is the area under the force–time graph. This is why crumple zones, airbags and bending your knees all work: the change in momentum is fixed by the collision, so extending the time reduces the force.

Conservation of momentum holds for any system with no external resultant force, and it holds in collisions and explosions alike. Kinetic energy is a different matter: it is conserved only in an elastic collision. In an inelastic collision momentum is still conserved while kinetic energy is not — it goes into deformation, heat and sound. In an explosion the total momentum stays zero if it started at zero, while kinetic energy increases from stored energy.