Revision: Laws of Motion JEE Main Laws of Motion

Aristotle's statement: “An external force is required to keep a body in uniform motion”.

The inability of a body to change its state of rest or of uniform motion or its direction by itself is called Inertia.

The inability of a body to change its state of uniform motion by itself is called inertia of motion.

The inability of a body to change its direction of motion by itself is called inertia of direction.

The inability of a body to change its state of rest by itself is called inertia of rest.

Newton’s second law of motion states that the rate of change of momentum is directly proportional to force applied and takes place in the direction of the force.

"Friction between two bodies in contact when one body is rolling over the other, is called rolling friction."

OR

The resistive force or rolling resistance that occurs when an object rolls across a surface and slows down the motion of a rolling ball/wheel, which is the weakest form of friction compared to static/sliding friction, is called Rolling Friction.

The force acting on a particle performing uniform circular motion along the radius and directed towards the centre of the circle is called the centripetal force.

The mathematical form of centripetal force is:

F = `mv^2/r`

where:

F = centripetal force,

m = mass of the object,

v = speed or velocity, and

r = radius

When a particle moves in two dimensions or in a plane such that its distance from a fixed (or moving) point remains constant, then its motion is called circular motion.

The circular motion in which the speed of a particle is constant but its direction changes continuously, and acceleration is always directed towards the centre, is called uniform circular motion.

The force directed along the radius towards the centre of a circle, which is necessary to keep the object moving in a circle, is called centripetal force.

The non-real (fictitious) force directed along the radius away from the centre of a circle (opposite to centripetal acceleration) is called centrifugal force.

Friction between two surfaces in contact when one body is actually sliding over the other body is called kinetic friction or dynamic friction.

OR

The force of friction that comes into play when a body is in a steady state of motion over another surface is called the force of kinetic friction.

OR

The resistive force that acts between moving surfaces that are in relative motion, always acting opposite to the direction of velocity and tending to slow down the speed of an object, expressed as F_k = μ_k × N, is called Kinetic Friction.

The angle between the resultant force (combination of normal force and frictional force) and the normal force itself, related to the coefficient of friction as tan⁡(ϕ) = μ, i.e., ϕ = tan⁡⁻¹(μ), is called the Angle of Friction.

The minimum angle of the rough inclined plane for which a body placed on it may just start sliding down, which is numerically equal to the angle of friction, is called the Angle of Repose.

The force which prevents or tries to prevent the slipping or sliding of two surfaces in contact, which can be high for dry and rough surfaces and low for smooth and wet surfaces, is called Friction.

\[\vec F\] = m \[\frac{d\vec{\mathrm{v}}}{dt}\] = m\[\vec a\] ... (for constant mass)

Thus, if \[\vec F\] = 0, \[\vec v\] is constant. Hence, if there is no force, velocity will not change. This is nothing but Newton's first law of motion.

General Form: \[\vec F\] =\[\frac{d\vec{p}}{dt}\]

For Constant Mass: \[\vec F\] = m\[\vec a\]

Momentum: \[\vec p\] = m\[\vec v\]

\[\vec{F}=\frac{d\vec{p}}{dt}=\frac{d\left(m\vec{\mathrm{v}}\right)}{dt}\]

\[\vec{F}=+\frac{mv^2}{r}\hat{r}_0\]

Directed away from the centre (positive sign indicates outward direction).

\[\vec{F}=-\frac{mv^2}{r}\hat{r}_0\]

Directed towards the centre (negative sign indicates inward direction).

μₖ = Fₖ/N

The coefficient of kinetic friction is defined as the ratio of force of kinetic friction to the normal reaction between the two surfaces in contact.

Fₖ = μₖ N

Where:

• Fₖ = Force of kinetic friction
• μₖ = Coefficient of kinetic friction (constant of proportionality)
• N = Normal reaction between the two surfaces in contact

Statement: The law of inertia shows that a body will preserve its velocity and direction till no force in the direction of its motion acts upon it.

• A more massive object has more inertia.
• To maintain uniform motion along a straight line, balanced forces are required.
• Unbalanced external forces acting on a body can only bring a change in its state of motion.

Statement:

Every inanimate object continues to be in a state of rest or of uniform unaccelerated motion along a straight line, unless it is acted upon by an external, unbalanced force.

Importance:

• It shows the equivalence between the state of rest and the state of uniform motion along a straight line — the distinction lies only in the choice of frame of reference.
• It defines force as a physical entity that brings about a change in the state of motion or rest of an object.
• It defines inertia as a fundamental and inherent property of every physical body by virtue of which it resists any change in its state of rest or uniform motion along a straight line.

Statement:

The rate of change of linear momentum of a rigid body is directly proportional to the applied (external unbalanced) force and takes place in the direction of force.

F = Δp = m\[\frac {dv}{dt}\] = ma

Importance:

• It provides a mathematical formulation for the quantitative measure of force: F = \[\frac {Δp}{Δt}\] = ma.
• It defines momentum as the product of mass and velocity: p = mv.
• Aristotle's fallacy is overcome by establishing that it is the resultant unbalanced force — not force itself — that is required to maintain a change in the state of motion.

Statement:

To every action (force) there is always an equal and opposite reaction (force).

Importance:

• It defines action and reaction as a pair of equal and opposite forces acting along the same line — whenever one object exerts a force on another, the second object exerts an equal and opposite force on the first.
• Action and reaction forces always act on different objects and therefore never cancel each other out.

Statement: The total momentum of a system of particles remains constant as long as no external forces act upon it.

• When no external forces act on colliding objects, the vector sum of linear momentum of each body remains constant and is not affected by mutual interaction.

Statement: When three forces F₁, F₂ and F₃ act on a body and are in equilibrium, each force is proportional to the sine of the angle between the other two forces.

Formula:

where α, β, and γ are the angles opposite to forces F₁, F₂, and F₃ respectively.

• First Law: When two bodies are in contact with each other, the direction of the force of friction on one at its point of contact is opposite to the direction in which the point of contact tends to move relative to the other.
• Second Law: When bodies are in equilibrium, the force of friction prevents the motion, which can be determined by using conditions of equilibrium of forces that act on the body.
• Third Law: The ratio of limiting friction to the normal reaction of surfaces depends on the nature of the substances and does not depend on the magnitude of the normal reaction.
• Fourth Law: The amount of limiting friction is independent of the area of contact or the shape of surfaces, provided the normal reaction remains unchanged.
• Fifth Law: During motion, the direction of friction is opposite to that of relative motion and is independent of velocity.

Statement: The initial momentum of a rocket at its launching pad is zero. When fired, the exhaust gases rush downward at high speed. To conserve momentum, the rocket moves upwards.

Thrust on the rocket:

The negative sign indicates that the direction of thrust is opposite to the direction of escaping gases.

Acceleration of the rocket:

where v = velocity of exhaust gases and \[\frac {dm}{dt}\]​ = rate of fuel consumption = rate of ejection of fuel.