Forces and motion in synthesis

Define system, frame and data

The system can be one body, several bodies or a body plus Earth. The choice determines which forces are external and where to account for potential energy. Select an approximately inertial frame, where the first law holds, and axes with positive directions. A free-body diagram includes only forces acting on the chosen body: weight, normal force, tension, friction or other identified interactions.

Velocity and acceleration are vectors. A body may have zero velocity at an instant and non-zero acceleration, as at the top of a vertical throw. It may retain speed while changing direction, as in uniform circular motion. Do not conclude zero resultant force just because speed stays constant. The second law relates resultant force to the change in the velocity vector, not only its magnitude.

Project the second law

For constant mass in the classical regime, ΣF=ma. In components, ΣFₓ=maₓ and ΣFᵧ=maᵧ. Mass is the same in any direction; force and acceleration components change. On an inclined plane at angle θ, axes parallel and perpendicular to the plane make weight easy to resolve: mg sinθ parallel downwards and mg cosθ perpendicular inwards.

Normal force is not automatically equal to weight. On an inclined plane without other perpendicular forces or perpendicular acceleration, N=mg cosθ. In an accelerating lift, normal force can be greater or smaller than mg. Kinetic friction has approximate magnitude μₖN and opposes relative sliding. Static friction adjusts up to a maximum μₛN; it does not always equal that maximum. The model must state which situation occurs.

Kinematics after dynamics

After finding acceleration, choose the motion description. If the component is constant, v=v₀+at and x=x₀+v₀t+at²/2. The relationship v²=v₀²+2aΔx eliminates time in one dimension. These formulas must not be used with variable acceleration without justification. In two dimensions, component independence allows time to be solved in one direction and used in the other.

In ideal projectile motion without air resistance and with constant g, aₓ=0 and aᵧ=−g for an upward-positive vertical axis. Horizontal velocity remains constant and vertical velocity varies. At the top vᵧ=0, but vₓ may be non-zero and acceleration remains downwards. The parabolic trajectory results from these two equations sharing the same time; it does not result from a horizontal force running out.

Work and energy: another entry point

The work of a constant force over a displacement is W=FΔr cosθ. The sum of work contributions is the change in kinetic energy K=mv²/2. This theorem connects states without calculating every intermediate instant. A force perpendicular to displacement does no work at that instant, even if it changes motion direction.

For conservative forces, we can write Wc=−ΔU. Near Earth’s surface, with approximately constant g, Ug=mgh relative to a chosen level. An ideal spring has Ue=kx²/2. If only these forces do work, K+U is conserved. With dissipative forces or other transfers, write Δ(K+U)=Wnc in the appropriate model. Friction can transform mechanical energy into internal energy; total energy remains conserved. Mechanical energy is part of the balance, not its entirety.

Circular motion needs no new force

On a circular arc of radius r, the radial acceleration component is v²/r towards the centre. If speed also changes, there is a tangential component. The radial equation is the sum of real force components towards the centre equal to mv²/r. The expression centripetal force names the required radial resultant; it must not be added to the diagram as an extra interaction.

On a horizontal bend, friction can provide the radial force. In a pendulum or vertical loop, weight and tension contribute with position-dependent signs. Energy can determine speed at a point, but only the force equation determines tension or normal force there. This combination is common: use energy to compare states and Newton to obtain an interaction. Check whether a calculated tension or normal force is physically possible, since ropes do not push and unilateral contacts do not pull.

Checking the result and its validity

A physical solution needs three types of check. Dimensional checking asks whether the equation produces the expected unit. Sign checking asks whether acceleration and work agree with the chosen direction. Limiting-case checking asks what happens if friction tends to zero, initial height is zero, or radius increases at constant speed. These tests can reveal a wrong formula before detailed calculation.

The examples use g=9,8 m/s² as an approximation and neglect effects where explicitly stated. Air resistance may matter for light objects or high speeds; friction may vary and the ground may deform. An exact answer within a simplified model must be presented as that model’s prediction. Choosing forces or energy depends on the question: forces reveal acceleration and interactions, energy connects states and transfers, and kinematics describes evolution when acceleration is known.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

Centripetal is an extra force besides weight and tension.

It names the radial resultant of real forces.

Normal force always equals mg.

It depends on geometry, other forces and perpendicular acceleration.

Mechanical energy is conserved in every situation.

Dissipation and work of non-conservative forces must be accounted for.