Laws of Motion

Master Newton's laws, friction, circular motion dynamics, and problem-solving techniques for JEE Physics.

Newton’s Laws of Motion form the foundation of classical mechanics. This chapter covers force, inertia, friction, and dynamics.

Overview

graph TD
    A[Laws of Motion] --> B[Newton's Laws]
    A --> C[Friction]
    A --> D[Circular Motion Dynamics]
    B --> B1[First Law - Inertia]
    B --> B2[Second Law - F=ma]
    B --> B3[Third Law - Action-Reaction]
    C --> C1[Static Friction]
    C --> C2[Kinetic Friction]
    D --> D1[Centripetal Force]
    D --> D2[Banking of Roads]

Newton’s First Law (Law of Inertia)

A body continues in its state of rest or uniform motion in a straight line unless acted upon by an external force.

Key Concepts:

  • Inertia: tendency to resist change in motion
  • Inertia depends on mass
  • Defines inertial reference frames

Newton’s Second Law

The rate of change of momentum is proportional to the applied force:

$$\boxed{\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a}}$$

For constant mass: $F = ma$

Impulse

$$\vec{J} = \int \vec{F} \, dt = \Delta \vec{p}$$

For constant force: $J = F \cdot \Delta t = m(v - u)$

JEE Tip
When solving problems, always draw a Free Body Diagram (FBD) first. Identify all forces acting on the body before applying Newton’s second law.

Newton’s Third Law

For every action, there is an equal and opposite reaction.

$$\vec{F}_{AB} = -\vec{F}_{BA}$$

Key Points:

  • Action and reaction act on DIFFERENT bodies
  • They are equal in magnitude, opposite in direction
  • They act simultaneously

Friction

Static Friction

$$f_s \leq \mu_s N$$
  • Opposes tendency of motion
  • Self-adjusting (up to maximum value)
  • Maximum static friction: $f_{s,max} = \mu_s N$

Kinetic Friction

$$f_k = \mu_k N$$
  • Opposes actual motion
  • Generally $\mu_k < \mu_s$
  • Independent of velocity (approximately)

Angle of Friction

$$\tan \theta = \mu$$

Angle of Repose

Minimum angle of incline at which body starts sliding:

$$\theta_R = \tan^{-1}(\mu_s)$$

Circular Motion Dynamics

Centripetal Force

Force required for circular motion:

$$\boxed{F_c = \frac{mv^2}{r} = m\omega^2 r}$$

This is not a new force but the resultant of existing forces directed toward center.

Vehicle on Level Road

Maximum safe speed:

$$v_{max} = \sqrt{\mu_s r g}$$

Vehicle on Banked Road

Without friction:

$$\tan \theta = \frac{v^2}{rg}$$

With friction:

$$v_{max} = \sqrt{rg \cdot \frac{\mu + \tan\theta}{1 - \mu\tan\theta}}$$ $$v_{min} = \sqrt{rg \cdot \frac{\tan\theta - \mu}{1 + \mu\tan\theta}}$$

Constraint Equations

For connected bodies, use:

  1. String constraint (inextensible string)
  2. Wedge constraint
  3. Pulley constraint

String Constraint: If string is inextensible, total length is constant.

Common Problem Types

1. Block on Incline

Forces: $mg\sin\theta$ (down the plane), $mg\cos\theta$ (into plane)

Without friction: $a = g\sin\theta$

With friction: $a = g(\sin\theta - \mu\cos\theta)$

2. Pulley Systems

Apply Newton’s second law to each block and use constraint equations.

3. Connected Bodies

Use FBD for each body and solve simultaneous equations.

Practice Problems

  1. A 5 kg block rests on a rough horizontal surface (μ = 0.4). Find the minimum force required to just move it.

  2. A car travels on a banked road of angle 30° and radius 100 m. Find the maximum safe speed if μ = 0.2.

  3. Two blocks of masses 5 kg and 3 kg are connected by a string over a pulley. Find the acceleration and tension.

Quick Check
Why is the reaction force on earth due to a falling apple not visible?