TheIBAxolotl
Forces and Newton's laws
A force is a push or pull that can change an object's velocity or shape. Newton's laws tie forces to motion:
- First law. If the resultant force on a body is zero, its velocity does not change: it stays at rest or keeps moving in a straight line at constant speed. Balanced forces mean no acceleration, not no motion.
- Second law. The resultant force equals the rate of change of momentum, \(F = \frac{\Delta p}{\Delta t}\). For constant mass this becomes \(F = ma\).
- Third law. When body A pushes on body B, B pushes back on A with a force equal in size, opposite in direction and of the same type. The pair acts on different bodies, so the two forces never cancel each other.
Free-body diagrams
Draw the object as a dot or box and show only the forces acting on it, each as an arrow starting at the object. Common forces:
- Weight \(W = mg\), toward the centre of the Earth.
- Normal force \(N\), perpendicular to a contact surface.
- Tension \(T\), along a rope or string, pulling away from the object.
- Friction, parallel to the surface, opposing sliding (or the tendency to slide). Static friction can take any value up to \(\mu_s N\); once sliding, kinetic friction is \(\mu_d N\), and usually \(\mu_d < \mu_s\).
- Buoyancy, equal to the weight of fluid displaced: \(F_b = \rho V g\).
- Drag, opposing motion through a fluid. For a small sphere at low speed, Stokes' law gives \(F_d = 6\pi\eta r v\).
- Elastic (spring) force, \(F_H = -kx\) (Hooke's law), valid up to the limit of proportionality.
Resolve forces into perpendicular components, usually along and across a slope or horizontal and vertical.
Momentum and impulse
Linear momentum is \(p = mv\), a vector in kg m s\(^{-1}\). Impulse is the change in momentum, \(J = F\Delta t = \Delta p\), and equals the area under a force–time graph. Air bags and crumple zones increase the collision time, so the same change of momentum needs a smaller average force.
In a closed system with no external resultant force, total momentum is conserved. This holds in every collision and explosion.
- Elastic collision: kinetic energy is also conserved.
- Inelastic collision: some kinetic energy becomes thermal energy, sound or deformation. If the bodies stick together it is perfectly inelastic.
- Explosion: a system initially at rest gains kinetic energy from stored energy, yet total momentum stays zero, so the fragments fly apart in opposite directions.
Circular motion
An object moving in a circle at constant speed is still accelerating, because its direction changes. The acceleration points toward the centre:
\[a = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2 r}{T^2}\]
The centripetal force \(F = \frac{mv^2}{r}\) is not a new kind of force. It is the name for the resultant inward force, supplied by friction (a car on a bend), tension (a ball on a string), gravity (an orbit) or a normal force. Angular speed is \(\omega = \frac{2\pi}{T}\) and \(v = \omega r\).
Worked example 1: friction and Newton's second law
A \(25\ \text{kg}\) crate is pushed across a level floor with a horizontal force of \(120\ \text{N}\). The coefficient of kinetic friction is \(0.30\). Find the acceleration (\(g = 9.81\ \text{m s}^{-2}\)).
- Normal force: \(N = mg = 25 \times 9.81 = 245\ \text{N}\)
- Friction: \(F_f = \mu_d N = 0.30 \times 245 = 73.6\ \text{N}\)
- Resultant: \(120 - 73.6 = 46.4\ \text{N}\)
- \(a = \frac{46.4}{25} = 1.9\ \text{m s}^{-2}\) (2 s.f.)
Worked example 2: a sticky collision
A \(2.0\ \text{kg}\) trolley at \(3.0\ \text{m s}^{-1}\) hits a stationary \(1.0\ \text{kg}\) trolley, and they move off together.
- Momentum before: \(2.0 \times 3.0 = 6.0\ \text{kg m s}^{-1}\)
- Velocity after: \(v = \frac{6.0}{3.0} = 2.0\ \text{m s}^{-1}\)
- Kinetic energy before: \(\frac{1}{2}(2.0)(3.0)^2 = 9.0\ \text{J}\); after: \(\frac{1}{2}(3.0)(2.0)^2 = 6.0\ \text{J}\)
- \(3.0\ \text{J}\) is transferred to other forms, so the collision is inelastic.
Common mistakes
- Drawing a third-law pair on the same free-body diagram as if they balanced.
- Adding a separate "centripetal force" arrow alongside the real forces.
- Assuming the normal force always equals \(mg\); it does not on a slope or when another force has a vertical component.
- Forgetting that momentum is a vector, so velocities in opposite directions need opposite signs.
- Claiming kinetic energy is conserved in every collision.
Exam tips
- Start every force problem with a labelled free-body diagram; it often earns a mark on its own.
- When asked to "state" a law, include the conditions (for example, "resultant external force is zero").
- For collisions, write the momentum equation first, then check kinetic energy only if the question asks about elasticity.
- In circular motion, identify which real force provides the centripetal force before writing any equation.
- Impulse questions with graphs: the area under the curve is the answer.