3.1 Momentum and Newton’s laws of motion

Mass is inertia: for a given resultant force, a larger mass produces a smaller acceleration through F=ma. It is a scalar measured in kilograms.

Keep mass separate from weight, which is mg, and use the chosen body’s mass in its equation of motion.

The same 10 N force gives 2 m s⁻² to a 5 kg object but 1 m s⁻² to a 10 kg object.

Mass does not depend on local g; an object’s weight can change while its inertia remains the same.

Newton’s second law states ΣF=ma. The acceleration vector has the same direction as the resultant and magnitude proportional to it for fixed mass.

Draw a free-body diagram, resolve components and include only forces acting on the chosen body. A negative component records the chosen sign convention.

A 3 kg trolley with resultant force (6,0) N accelerates at (2,0) m s⁻².

Using one applied force instead of the resultant gives the wrong acceleration when several forces act.

Momentum p=mv is a vector in the direction of velocity, measured in kg m s⁻¹. It is conserved for an isolated system when external impulse is negligible.

Use signed velocities on one axis and define the system before writing conservation. Momentum is distinct from kinetic energy, which depends on speed squared.

A 2 kg object moving at −3 m s⁻¹ has momentum −6 kg m s⁻¹ in the chosen positive direction.

Equal and opposite momenta can sum to zero while individual objects still move.

The general law is F=dp/dt. For constant mass it becomes F=ma, but variable-mass systems require momentum flux to be considered explicitly.

Choose the system boundary and direction before differentiating momentum. The force is the resultant external force on that system.

A constant 12 N force acting on a 3 kg object produces dp/dt=12 kg m s⁻² and, at fixed mass, acceleration 4 m s⁻².

F=ma is a special constant-mass form; force is not simply momentum divided by time unless the change is defined correctly.

First law defines inertial motion when resultant force is zero; second law gives ΣF=ma; third law pairs equal opposite forces acting on different bodies.

Apply one law to one chosen body at a time. Identify the frame, draw forces and distinguish acceleration from velocity.

A stationary book has weight and normal reaction balanced; the Earth–book gravitational interaction has a third-law partner acting on Earth.

Third-law forces do not cancel in one free-body diagram because they act on different objects.

Weight W=mg is the force exerted by a gravitational field on mass m. Its direction is toward the field source and its value depends on local g.

Use mass in inertial equations and weight as a force in free-body diagrams. In a non-uniform field, g can vary with position.

A 5 kg mass near Earth has weight about 49 N downward when g=9.8 m s⁻².

Weight is not mass, and apparent weight measured by a scale can differ from mg during acceleration.