Class 11 Physics · Chapter 4 NotesLaws of Motion

Aristotle believed that an external force is needed to keep a body in motion. This seems true in daily life because a toy car stops when you stop pulling it, but the real reason is the opposing force of friction. Galileo studied balls on inclined planes and found that a ball moving down accelerates, one moving up retards, and on a horizontal plane it should move with constant velocity if friction were absent. From a double inclined plane experiment, he concluded that a ball released from a certain height always rises to nearly the same height, and on a horizontal plane it would travel forever. He thus arrived at the law of inertia: the state of rest and the state of uniform linear motion are equivalent, and no net force is needed to maintain uniform motion.

• Aristotle's law of motionan external force is required to keep a body in motion
• Galileo's law of inertiaa body moves with constant velocity on a frictionless horizontal plane
• Friction is the reason bodies in everyday life come to rest

Newton rephrased Galileo's law of inertia as the 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 simple terms, if the net external force on a body is zero, its acceleration is zero. This law introduces the concept of inertia, which means resistance to change of state. A book resting on a table stays at rest because the normal force R balances its weight W, not because the forces cancel and cause rest. When a bus starts suddenly, we are thrown backward because our feet move with the bus due to friction while the upper body tends to remain at rest due to inertia.

• First lawif net external force on a body is zero, its acceleration is zero
• Inertia means resistance to change of state of rest or uniform motion
• Mass is a measure of inertia

Momentum of a body is the product of its mass and velocity, p = mv, and is a vector quantity. Newton's 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 in which the force acts. In SI units we write F = dp/dt = ma, where F is the net external force. The law is a vector equation, so it holds separately for the x, y and z components. It applies to a single particle, and also to a rigid body or a system of particles if F is the total external force and a is the acceleration of the system as a whole. The SI unit of force is the newton: 1 N = 1 kg m s⁻².

• Momentump = mv
• Second lawF = dp/dt = ma
• 1 N = 1 kg m s⁻²
• Force changes only the component of velocity along its direction

When a large force acts for a very short time and produces a finite change in momentum, the product of force and time is called impulse. Impulse = Force × time duration = change in momentum. For example, when a ball hits a wall and bounces back, the force is large but acts for a very short time, so it is difficult to measure the force and time separately. However, the impulse, which equals the change in momentum, is easy to calculate. A large force acting for a short time to produce a finite change in momentum is called an impulsive force. In Newtonian mechanics, an impulsive force is like any other force, only large and short-lived.

• Impulse = F × t = change in momentum
• Impulse is a vector in the direction of the force
• Useful when force and time of contact are not known separately

Newton's third law states that to every action there is always an equal and opposite reaction. In simple terms, 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, that is F_AB = –F_BA. Action and reaction act on different bodies, act at the same instant, and there is no cause-effect relation between them. From the second and third laws, we get the law of conservation of momentum: the total momentum of an isolated system of interacting particles is conserved. This explains the recoil of a gun when a bullet is fired, and applies to all collisions, whether elastic or inelastic.

• Third lawF_AB = –F_BA
• Action and reaction act on different bodies and are simultaneous
• Law of conservation of momentumtotal momentum of an isolated system is constant

A particle is in equilibrium when the net external force on it is zero, so it is either at rest or in uniform motion. For two forces, equilibrium requires F₁ = –F₂. For three concurrent forces, the vector sum must be zero, and the forces can be represented by the sides of a triangle taken in the same sense. Common forces in mechanics include the gravitational force, which acts at a distance, and contact forces such as the normal reaction, friction, tension in a string, spring force F = –kx, buoyant force and viscous drag. All contact forces arise from electrical forces between the charged constituents of bodies, though at the macroscopic level they are treated as separate types of forces.

• Equilibriumnet external force on the particle is zero
• Normal reaction is perpendicular to the surfaces in contact
• Spring forceF = –kx, where k is the force constant
• Tension is the restoring force in a string

Friction is the component of the contact force parallel to the surfaces in contact, and it opposes impending or actual relative motion. Static friction acts when the body is at rest and adjusts itself up to a maximum value given by (fs)max = μsN, so fs ≤ μsN. Once the applied force exceeds this limit, the body begins to slide and kinetic friction acts, given by fk = μkN. The coefficients μs and μk depend only on the nature of the surfaces in contact, and μk is less than μs. Friction is independent of the area of contact. Rolling friction is much smaller than static or sliding friction, which is why ball bearings and air cushions are used to reduce friction in machines.

• Static frictionfs ≤ μsN
• Kinetic frictionfk = μkN
• μk < μs
• Friction is independent of the area of contact

For a body moving in a circle of radius R with uniform speed v, the acceleration v²/R is directed towards the centre, and the force providing it is called the centripetal force, fc = mv²/R. For a car on a level road, this force is provided by friction, and the maximum safe speed is v_max = √(μsRg). On a banked road, the optimum speed with no friction needed is v₀ = √(Rg tan θ), and the maximum permissible speed is greater than on a flat road. To solve problems in mechanics, draw a free-body diagram of the chosen system, include all external forces on it, and apply Newton's laws. Internal forces between parts of the system cancel in pairs.

• Centripetal forcefc = mv²/R
• Level roadv_max = √(μsRg)
• Banked road optimum speedv₀ = √(Rg tan θ)
• Free-body diagram shows only external forces on the chosen system