Laws of Motion

The Three Laws of Classical Mechanics:

First Law (Law of Inertia):

Every physical body continues in its state of rest or uniform motion in a straight line unless acted upon by a non-zero external resultant force. Inertia is the inherent resistance of matter to any alteration in its state of motion; mass is the quantitative measure of inertia.

Second Law (Fundamental Law of Dynamics):

The time rate of change of linear momentum of a body is directly proportional to the applied net force and takes place in the direction in which the force acts: $$\vec{F} = \frac{d\vec{p}}{dt}$$ Since momentum is $\vec{p} = m\vec{v}$, for constant mass $m$: $$\vec{F} = \frac{d(m\vec{v})}{dt} = m\frac{d\vec{v}}{dt} = m\vec{a}$$ Note: Newton's Second Law is the most fundamental law of mechanics because both the First and Third Laws can be mathematically deduced from it.

Third Law (Action and Reaction):

To every action, there is an equal and opposite reaction: $$\vec{F}_{12} = -\vec{F}_{21}$$ Key Fact: Action and reaction forces act simultaneously on TWO DIFFERENT bodies. Hence, they never form an equilibrium pair and never cancel each other!

Impulse Definition & Mathematical Formulation:

Impulse ($\vec{J}$) is a vector quantity measuring the total effect of a force acting over a finite time interval $\Delta t$: $$\vec{J} = \int_{t_1}^{t_2} \vec{F}\, dt = \vec{F}_{\text{avg}} \cdot \Delta t$$

Impulse-Momentum Theorem:

From $\vec{F} = \frac{d\vec{p}}{dt} \implies \vec{F}\, dt = d\vec{p}$. Integrating: $$\vec{J} = \int_{\vec{p}_i}^{\vec{p}_f} d\vec{p} = \vec{p}_f - \vec{p}_i = \Delta\vec{p}$$ The impulse of a force equals the change in linear momentum of the body.

Real-World Applications:

• Catching a Cricket Ball: A fielder pulls his hands backward while catching a fast ball. Increasing the impact time $\Delta t$ drastically reduces the peak impulsive force $F = \frac{\Delta p}{\Delta t}$, preventing hand injury.
• Automobile Airbags & Crumple Zones: Vehicles are engineered to crumple during collisions, prolonging the deceleration time of passengers to minimize traumatic impact forces.
• Shock Absorbers: Dampen impulsive road vibrations by distributing the momentum transfer over extended time intervals.

Principle of Conservation of Linear Momentum:

In the absence of any external net force ($\vec{F}_{\text{ext}} = 0$): $$\frac{d\vec{P}}{dt} = 0 \implies \mathbf{\vec{P} = \sum m_i \vec{v}_i = \text{constant}}$$ Total linear momentum of an isolated system is conserved in all frames of reference.

Classic Application: Recoil of a Gun

Before firing, total momentum is zero. After firing a bullet of mass $m$ with muzzle velocity $\vec{v}$, the gun of mass $M$ recoils with velocity $\vec{V}$: $$m\vec{v} + M\vec{V} = 0 \implies \mathbf{\vec{V} = -\frac{m}{M}\vec{v}}$$ The negative sign indicates recoil opposite to bullet direction. Because $M \gg m$, recoil speed $|V| \ll |v|$.

Variable Mass System: Rocket Propulsion

A rocket exhausts fuel gases at constant relative exhaust speed $u$ at a mass consumption rate $\frac{dm}{dt}$. The upward thrust force is: $$F_{\text{thrust}} = u \left(-\frac{dm}{dt}\right)$$ Integrating the equation of motion yields Tsiolkovsky's rocket equation: $$v(t) = v_0 + u \ln\left(\frac{m_0}{m}\right) - gt$$

Equilibrium of Concurrent Forces:

A system of forces acting on a single particle is concurrent. The particle is in translational equilibrium if the vector sum of all concurrent forces is zero: $$\sum \vec{F} = 0 \implies \sum F_x = 0, \quad \sum F_y = 0, \quad \sum F_z = 0$$

Lami's Theorem:

If three concurrent, coplanar forces $\vec{F}_1, \vec{F}_2, \vec{F}_3$ acting on a particle maintain it in static equilibrium, then each force magnitude is directly proportional to the sine of the angle between the other two forces: $$\frac{F_1}{\sin\alpha} = \frac{F_2}{\sin\beta} = \frac{F_3}{\sin\gamma}$$ where $\alpha$ is the angle between $\vec{F}_2$ and $\vec{F}_3$, $\beta$ between $\vec{F}_1$ and $\vec{F}_3$, and $\gamma$ between $\vec{F}_1$ and $\vec{F}_2$.

Microscopic Origin and Nature of Friction:

Friction is a contact force acting tangentially at the interface of two surfaces in contact, opposing relative motion or the tendency of relative motion. Microscopically, it originates from electromagnetic forces and cold welding between microscopic surface asperities.

1. Angle of Friction ($\lambda$):

The angle of friction is the angle which the resultant ($\vec{S}$) of limiting friction ($\vec{f}_L$) and normal reaction ($\vec{N}$) makes with the normal reaction: $$\tan\lambda = \frac{f_L}{N} = \frac{\mu_s N}{N} \implies \mathbf{\tan\lambda = \mu_s}$$

2. Angle of Repose ($\theta$):

The angle of repose is the maximum inclination angle of an inclined plane at which a body placed upon it rests in limiting equilibrium without sliding down under gravity:

• Down-plane component of gravity: $F_{\text{down}} = mg\sin\theta$
• Normal reaction: $N = mg\cos\theta$
• At limiting equilibrium: $mg\sin\theta = f_L = \mu_s N = \mu_s mg\cos\theta$
• Dividing gives: $\tan\theta = \mu_s \implies \mathbf{\theta = \tan^{-1}\mu_s}$

Vehicular Dynamics on Curved Roads:

Case 1: Level (Unbanked) Curved Road of Radius $r$

Centripetal force is provided solely by static friction between tires and road: $$f_s = \frac{mv^2}{r} \le \mu_s N = \mu_s mg \implies \mathbf{v_{\max} = \sqrt{\mu_s r g}}$$ Exceeding $v_{\max}$ results in outward skidding.

Case 2: Banked Road Without Friction (Optimum Speed)

The road is elevated at angle $\theta$. The horizontal component of normal force $N\sin\theta$ provides centripetal acceleration: $$N\sin\theta = \frac{mv^2}{r}, \quad N\cos\theta = mg \implies \tan\theta = \frac{v^2}{rg} \implies \mathbf{v_0 = \sqrt{rg\tan\theta}}$$ At this exact speed, there is zero lateral friction and zero tire wear.

Case 3: Banked Road With Friction (Maximum Safe Speed)

Resolving normal force $N$ and limiting friction $f_L = \mu_s N$: $$\mathbf{v_{\max} = \sqrt{rg\left(\frac{\mu_s + \tan\theta}{1 - \mu_s\tan\theta}\right)}}$$ $$\mathbf{v_{\min} = \sqrt{rg\left(\frac{\tan\theta - \mu_s}{1 + \mu_s\tan\theta}\right)}}$$

Standard Board Free Body Diagram (FBD) Problems: