Chapter 4: Laws of Motion | Summary

Galileo studied motion on inclined planes and concluded that an object on a frictionless horizontal surface would move with constant velocity forever, since it experiences neither acceleration nor retardation. His double inclined plane experiment showed that a ball released from one slope always rises to the same height on the other. When the second slope is made horizontal, the ball would travel an infinite distance.

From these observations, Galileo arrived at the law of inertia: a body at rest remains at rest and a body in uniform motion continues in uniform motion unless acted upon by a net external force. Inertia means resistance to change. Both rest and uniform linear motion correspond to zero net force and zero acceleration.

Newton formalised Galileo's law of inertia as his First Law of Motion: every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise. In simpler terms, if the net external force on a body is zero, its acceleration is zero.

The First Law serves two practical purposes. If we know the net external force is zero, we conclude the acceleration is zero. Conversely, if we observe a body at rest or in uniform motion, we conclude the net external force on it must be zero. For example, a book resting on a table has its weight balanced by the normal reaction, giving zero net force.

The Second Law states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. Momentum p is defined as the product of mass m and velocity v: p = mv. The mathematical form is F = dp/dt = ma, where the SI unit of force is the newton (1 N = 1 kg m s⁻²).

Key properties of the Second Law: it is consistent with the First Law (F = 0 implies a = 0); it is a vector law equivalent to three component equations; it applies to a particle, rigid body, or system of particles when F represents total external force and a is the acceleration of the system as a whole; and it is a local law—acceleration at an instant depends only on force at that instant, not on the history of motion. Impulse, defined as force multiplied by time duration, equals the change in momentum and is useful when a large force acts for a very short time.

Newton's Third Law states: to every action there is always an equal and opposite reaction. Forces always occur in pairs; the force on body A by body B is equal and opposite to the force on body B by A. Importantly, action and reaction forces act on different bodies simultaneously—there is no cause-effect relationship, and either force may be called the action.

A common misconception is that action-reaction forces cancel each other. They do not, because they act on different bodies. However, when treating a system of two bodies together, these mutual forces become internal forces that cancel in pairs, enabling the Second Law to apply to the system as a whole. The Third Law holds even when bodies are not in contact, as in gravitational interactions.

The law of conservation of momentum follows directly from the Second and Third Laws: the total momentum of an isolated system (no external force) is conserved. When a bullet is fired from a gun, the momentum gained by the bullet is equal and opposite to the recoil momentum of the gun, so total momentum remains zero.

For a collision between two bodies A and B, applying the Second Law to each and using the Third Law (FAB = −FBA) shows that the total final momentum equals the total initial momentum. This holds for both elastic and inelastic collisions. In elastic collisions, an additional condition requires that total kinetic energy is also conserved.

A particle is in equilibrium when the net external force on it is zero, meaning it is either at rest or in uniform motion. For two forces, equilibrium requires them to be equal and opposite. For three concurrent forces F₁, F₂, and F₃, equilibrium requires their vector sum to be zero: F₁ + F₂ + F₃ = 0.

Graphically, three forces in equilibrium can be represented by the sides of a closed triangle with arrows in the same sense. This extends to n forces represented by a closed n-sided polygon. In component form, the sum of each x, y, and z component of all forces must independently equal zero. Free-body diagrams are a systematic tool for identifying all forces on a chosen system.

The gravitational force is pervasive and can act at a distance. All other common forces in mechanics are contact forces, arising due to contact between solid or fluid bodies. Contact forces include normal reaction (component perpendicular to surfaces in contact), friction (component parallel to surfaces), buoyant force, viscous drag, tension in strings, and spring force. Spring force obeys Hooke's law: F = −kx, where k is the force constant and x is displacement from the equilibrium position.

At the microscopic level, all contact forces fundamentally arise from electrical forces between charged constituents (nuclei and electrons) of matter. However, for practical mechanics at the macroscopic scale, they are treated as distinct types of forces with empirically determined properties. Only gravitational and electrical forces are relevant in classical mechanics; weak and strong nuclear forces operate at subatomic scales.

Static friction (fs) opposes impending relative motion and is self-adjusting up to a maximum value: (fs)max = μsN, where μs is the coefficient of static friction and N is the normal force. The law of static friction is fs ≤ μsN. Static friction does not exist without an applied force and is independent of the area of contact.

Kinetic (sliding) friction (fk) opposes actual relative motion and satisfies fk = μkN, where μk is the coefficient of kinetic friction. Experimentally, μk < μs. Rolling friction is much smaller than static or sliding friction because at every instant only one point of the rolling body contacts the surface with zero relative motion. Friction, while sometimes undesirable in machines, is essential for walking, driving, and braking. Lubricants and ball bearings reduce kinetic friction in machines.

A body moving in a circle of radius R with uniform speed v has centripetal acceleration v²/R directed towards the centre. The centripetal force providing this acceleration is fc = mv²/R. This is not a new type of force but a name for whichever material force (tension, gravity, friction, normal reaction) acts centripetally.

For a car on a level road taking a circular turn, static friction provides the centripetal force. The maximum speed before slipping is vmax = √(μsRg). For a car on a banked road at angle θ, the horizontal components of normal reaction and friction together provide the centripetal force. The optimum speed at which no friction is needed is v₀ = √(Rg tanθ). The maximum permissible speed on a banked road is higher than on a flat road.

A systematic approach to mechanics problems involves: drawing a schematic diagram of the assembly; choosing a convenient system; drawing a free-body diagram showing all external forces on the chosen system (excluding forces exerted by the system on the environment); and applying Newton's laws to determine unknowns. Internal forces within a system do not appear in its free-body diagram.

When applying the Third Law across two sub-systems, if the force on A due to B is F in A's free-body diagram, then the force on B due to A must be −F in B's free-body diagram. Free-body diagrams help in correctly identifying action-reaction pairs: only mutual forces between two different bodies constitute such pairs; two forces on the same body that happen to be equal and opposite do not.

Chapter 4 establishes Newton's three laws of motion as the foundation of classical mechanics. The First Law defines inertia and the concept of force; the Second Law provides a quantitative relationship between force, mass, and acceleration through the concept of momentum; and the Third Law reveals that forces always occur in pairs acting on different bodies. Together, these laws lead to the principle of conservation of momentum, one of the most powerful tools in physics. The treatment of friction, equilibrium, and circular motion demonstrates how these laws apply to a wide range of real-world situations.

For board examinations, students must be able to state all three laws precisely, apply F = ma and impulse-momentum theorem to numerical problems, distinguish between static and kinetic friction, draw and interpret free-body diagrams, and derive the maximum speed conditions for circular motion on level and banked roads. Questions on action-reaction pairs, conservation of momentum in collisions, and Aristotle's fallacy are frequently tested. Mastery of free-body diagrams is the single most important skill for solving complex multi-body problems in this and subsequent chapters.