Laws of Motion

For nearly two millennia, natural philosophy was dominated by the Aristotelian view: "An external force is continuously required to keep a body in uniform motion." Aristotle reasoned from everyday observation where a rolling cart stops unless pulled.

Galileo Galilei refuted this fallacy through his famous double inclined plane experiments:

• A ball released from a height on a smooth downward slope rolls up an opposing slope to reach nearly the same vertical height.
• When the slope of the second incline is decreased, the ball travels a greater distance to reach the same height.
• In the limiting case where the second plane is completely horizontal, the ball must travel an infinite distance with constant speed to attain its initial height!

Galileo concluded that uniform motion in a straight line is as natural as a state of rest; a body stops solely due to opposing resistive forces such as friction and air resistance. In the complete absence of opposing forces, no external force is needed to sustain uniform rectilinear motion.

Inertia is the inherent property of a material body by virtue of which it resists any change in its state of rest or of uniform motion along a straight line.

• Inertia of Rest: Tendency of a body to remain at rest. Examples: Passengers jerk backward when a bus starts abruptly; shaking a tree branch causes ripe fruit to detach.
• Inertia of Motion: Tendency of a body to persist in uniform motion. Examples: Passengers lurch forward when brakes are applied suddenly; an athlete runs a distance before leaping in long jump.
• Inertia of Direction: Inability of a body to alter its direction of travel without an external force. Examples: Mud flies off rotating tires tangentially; passengers lean outward when a vehicle rounds a sharp curve.
• Quantitative Measure: Mass ($m$) is the quantitative measure of inertia. A more massive body requires a larger force to achieve a given acceleration ($I \propto m$).

Sir Isaac Newton formalized Galileo's discovery into his First Law of Motion:

The First Law provides two vital physical insights:

1. Qualitative Definition of Force: Force is an external agency (push or pull) that changes or tends to change the state of rest or uniform motion of a body.
2. Definition of Inertial Reference Frames: A reference frame where the First Law holds strictly is an Inertial Frame. Any frame moving at constant velocity relative to an inertial frame is also inertial.

A reference frame that undergoes acceleration $ec{a}_0$ or rotation relative to an inertial frame is a Non-Inertial Frame. In such frames, Newton's laws cannot be applied directly without introducing a fictitious or Pseudo-Force ($ec{F}_{ ext{pseudo}}$):

where $m$ is the mass of the body and $ec{a}_0$ is the acceleration of the observer's frame. The pseudo-force acts in the direction strictly opposite to the frame's acceleration.

A heavy cricket ball moving slowly and a light bullet moving at high speed both possess immense capability to produce damage upon impact. The combined effect of mass and motion is quantified by Linear Momentum:

• Linear momentum is a vector quantity possessing the same direction as velocity $ec{v}$.
• SI Unit: $ ext{kg}\cdot ext{m/s}$ or $ ext{N}\cdot ext{s}$; CGS Unit: $ ext{g}\cdot ext{cm/s}$.
• Dimensional Formula: $[M L T^{-1}]$.

Mathematically: $$ec{F} \propto rac{dec{p}}{dt} \implies ec{F} = krac{dec{p}}{dt}$$ Choosing the unit of force such that $k = 1$: $$ec{F} = rac{dec{p}}{dt} = rac{d(mec{v})}{dt} = mrac{dec{v}}{dt} + ec{v}rac{dm}{dt}$$

For systems of constant mass ($rac{dm}{dt} = 0$), this reduces to the celebrated equation: $$\mathbf{ec{F} = mec{a}}$$

In Cartesian components: $$F_x = m a_x = mrac{d^2x}{dt^2}, \quad F_y = m a_y = mrac{d^2y}{dt^2}, \quad F_z = m a_z = mrac{d^2z}{dt^2}$$

When a large force acts for an extremely short time interval (such as a bat hitting a ball, or a hammer striking a nail), it is termed an impulsive force. The overall effect of such a force is measured by Impulse ($ec{J}$):

From Newton's Second Law, $ec{F} = rac{dec{p}}{dt} \implies ec{F}\,dt = dec{p}$. Integrating between limits gives the Impulse-Momentum Theorem: $$\mathbf{ec{J} = \Deltaec{p} = ec{p}_2 - ec{p}_1 = mec{v}_2 - mec{v}_1}$$

• Impulse equals the change in linear momentum of the body.
• On a Force-time ($F ext{--}t$) graph, Area under the $F ext{--}t$ curve = Impulse = $\Delta p$.
• Practical Applications of $F_{ ext{avg}} = rac{\Delta p}{\Delta t}$:
  • A cricket fielder pulls their hands backward while catching a fast ball: increasing $\Delta t$ significantly reduces the average impact force $F_{ ext{avg}}$ exerted on the hands.
  • Automotive airbags and crumple zones increase deceleration time during collisions, minimizing injury.
  • Shock absorbers in motorcycles and railway buffers absorb kinetic impulses over larger time windows.

In a rocket, mass is not constant because fuel is continuously consumed and ejected as exhaust gases with a high relative speed $v_{ ext{rel}}$ at rate $-rac{dm}{dt}$.

• Upward Thrust Force on Rocket: $$F_{ ext{thrust}} = -v_{ ext{rel}}rac{dm}{dt} \quad ( ext{since } dm/dt < 0)$$
• Net Upward Acceleration at Time $t$: $$a = rac{F_{ ext{thrust}} - mg}{m} = rac{v_{ ext{rel}}}{m}\left(-rac{dm}{dt} ight) - g$$
• Tsiolkovsky Rocket Equation (Velocity at burnout): $$v = u + v_{ ext{rel}}\ln\left(rac{m_0}{m} ight) - gt$$ where $m_0$ is the initial launch mass and $m$ is the instantaneous mass.

where $ec{F}_{AB}$ is the force exerted on body $A$ by body $B$, and $ec{F}_{BA}$ is the force exerted on body $B$ by body $A$.

Crucial Conceptual Pillars of the Third Law:

• Forces in nature always occur in matched pairs; an isolated single force cannot exist.
• Action and reaction act simultaneously; neither is the cause nor the effect of the other.
• Action and reaction act on two DIFFERENT bodies. Consequently, they can NEVER cancel each other out to produce equilibrium! Equilibrium occurs only when equal and opposite forces act on the same body.

Newton's Second Law is called the Real Law of Motion because both the First Law and the Third Law are mathematically contained within it.

1. Second Law Contains First Law:

From the Second Law, $ec{F} = mec{a}$. If the net external force acting on a body is zero ($ec{F} = ec{0}$): $$mec{a} = ec{0} \implies ec{a} = ec{0} \quad ( ext{since mass } m eq 0)$$ $$rac{dec{v}}{dt} = ec{0} \implies ec{v} = ext{constant}$$ This means that if a body is at rest ($ec{v} = ec{0}$), it remains at rest; if moving with constant velocity, it maintains that velocity along a straight line in the absence of an external force. This is precisely Newton's First Law!

2. Second Law Contains Third Law:

Consider an isolated system consisting of two interacting bodies $A$ and $B$, free from any external forces ($\sumec{F}_{ ext{ext}} = ec{0}$).

Let $ec{F}_{AB}$ be the force exerted on $A$ by $B$, producing rate of change of momentum $rac{dec{p}_A}{dt}$.

Let $ec{F}_{BA}$ be the force exerted on $B$ by $A$, producing rate of change of momentum $rac{dec{p}_B}{dt}$.

According to the Second Law, total force acting on the isolated system is: $$ec{F}_{ ext{net}} = rac{d}{dt}(ec{p}_A + ec{p}_B) = ec{F}_{AB} + ec{F}_{BA}$$ Since no external force acts on the isolated system, $ec{F}_{ ext{net}} = ec{0}$: $$ec{F}_{AB} + ec{F}_{BA} = ec{0} \implies \mathbf{ec{F}_{AB} = -ec{F}_{BA}}$$ This proves Newton's Third Law!

Classic Physical Applications:

• Recoil of a Gun: Before firing, gun of mass $M$ and bullet of mass $m$ are at rest ($ec{p}_{ ext{initial}} = ec{0}$). Upon firing, bullet moves with muzzle velocity $ec{v}$ and rifle recoils with velocity $ec{V}$: $$mec{v} + Mec{V} = ec{0} \implies \mathbf{ec{V} = -rac{m}{M}ec{v}}$$ The recoil speed is inversely proportional to the gun's mass ($V \ll v$), explaining why heavy firearms kick less.
• Explosion of a Bomb at Rest: If a stationary bomb of mass $M$ explodes into two fragments of masses $m_1$ and $m_2$ with velocities $ec{v}_1$ and $ec{v}_2$: $$m_1ec{v}_1 + m_2ec{v}_2 = ec{0} \implies m_1ec{v}_1 = -m_2ec{v}_2$$ The fragments fly apart in exactly opposite directions with speeds inversely proportional to their masses.

Forces whose lines of action pass through a single common point are called Concurrent Forces. A body under the action of concurrent forces is in translational equilibrium if and only if the vector sum of all forces is zero:

Lami's Theorem: If three coplanar concurrent forces $ec{P}, ec{Q}, ec{R}$ acting on a body keep it in equilibrium, each force is proportional to the sine of the angle between the other two forces:

where $lpha$ is the angle between $ec{Q}$ and $ec{R}$, $eta$ is between $ec{P}$ and $ec{R}$, and $\gamma$ is between $ec{P}$ and $ec{Q}$.

A Free Body Diagram (FBD) is a diagrammatic representation where the body of interest is isolated from its environment, and all external contact forces (normal reaction $N$, tension $T$, friction $f$, push/pull) and field forces (gravity $mg$) acting ON the body are drawn as vectors originating from its center of mass.

Consider two masses $m_1$ and $m_2$ ($m_1 > m_2$) connected by a light, inextensible string passing over a frictionless, massless pulley:

• FBD of $m_1$: $m_1 g - T = m_1 a$
• FBD of $m_2$: $T - m_2 g = m_2 a$
• Adding equations gives the common acceleration ($a$): $$\mathbf{a = \left(rac{m_1 - m_2}{m_1 + m_2} ight)g}$$
• Eliminating $a$ gives the string tension ($T$): $$\mathbf{T = \left(rac{2m_1 m_2}{m_1 + m_2} ight)g}$$
• Thrust force on the pulley axle: $F_{ ext{pulley}} = 2T = rac{4m_1 m_2}{m_1 + m_2}g$.

A person of mass $m$ stands on a weighing machine placed inside an elevator. The machine registers the normal reaction $R$ exerted on the person:

Friction is an opposing tangential contact force that arises between two surfaces whenever there is relative motion or a tendency of relative motion between them.

• Classical View: Interlocking of microscopic surface irregularities (asperities).
• Modern Molecular View: At microscopic scales, actual contact occurs only at high points (junctions) comprising a tiny fraction ($< 0.01\%$) of apparent area. Tremendous local pressure causes cold-welding adhesion. Friction is the force needed to shear these microscopic cold-welded intermolecular bonds.

• Static Friction ($f_s$): Opposes the tendency of relative motion when surfaces are at rest. It is a self-adjusting force that automatically balances the applied force $F_{ ext{applied}}$: $$0 \le f_s \le f_L$$
• Limiting Friction ($f_L$): The maximum value of static friction just before sliding begins.
• Kinetic (Dynamic) Friction ($f_k$): Opposes actual sliding motion once surfaces are moving relative to each other ($f_k < f_L$).
• Rolling Friction ($f_r$): Resistance encountered when a sphere or cylinder rolls over a surface. Due to surface deformation, $f_r \ll f_k \ll f_L$, which explains why wheels and ball bearings revolutionize transportation.

Laws of Limiting Friction:

1. Limiting friction is directly proportional to normal reaction: $\mathbf{f_L = \mu_s N}$.
2. Independent of the apparent area of contact (as long as normal force is constant).
3. Depends strictly on the nature, material, and roughness of contacting surfaces.

Similarly, for kinetic friction: $\mathbf{f_k = \mu_k N}$, where $\mu_k < \mu_s$. Both $\mu_s$ and $\mu_k$ are dimensionless constants ($[M^0 L^0 T^0]$).

For an incline inclined at angle $ heta$ to the horizontal:

• Normal reaction: $N = mg\cos heta$.
• Gravitational component down the incline: $mg\sin heta$.
• Case 1: When $ heta > ext{Angle of Repose}$ (Body slides down freely): $$mg\sin heta - f_k = ma \implies mg\sin heta - \mu_k mg\cos heta = ma$$ $$\mathbf{a = g(\sin heta - \mu_k\cos heta)}$$
• Case 2: Minimum force to drag body UP the incline: $$F_{ ext{up}} = mg\sin heta + f_L = mg\sin heta + \mu_s mg\cos heta = \mathbf{mg(\sin heta + \mu_s\cos heta)}$$

As established in kinematics, a body of mass $m$ executing uniform circular motion of radius $r$ at constant speed $v$ experiences an inward radial centripetal acceleration $a_c = v^2/r$.

By Newton's Second Law, this requires a net inward radial force called the Centripetal Force ($F_c$):

When a vehicle of mass $m$ turns on a flat circular track of radius $r$ at speed $v$, the only horizontal force available to provide centripetal acceleration is the static friction ($f_s$) between the tires and the asphalt:

Therefore, the Maximum Safe Speed on a Level Road is: $$\mathbf{v_{\max} = \sqrt{\mu_s r g}}$$ If $v > v_{\max}$, available friction is insufficient, and the car skids radially outward!

Relying solely on friction is dangerous because friction decreases drastically in rain, ice, or tire wear. To eliminate reliance on friction, roads and railway tracks are banked: the outer edge is raised above the inner edge by an angle $ heta$.

Case 1: Frictionless Banking (Ideal / Design Speed $v_{ ext{opt}}$):

The normal reaction $N$ tilts inward at angle $ heta$ to the vertical:

• Vertical equilibrium: $N\cos heta = mg$
• Horizontal centripetal force: $N\sin heta = rac{mv^2}{r}$
• Dividing the two equations: $$rac{N\sin heta}{N\cos heta} = rac{mv^2/r}{mg} \implies \mathbf{ an heta = rac{v^2}{rg}}$$
• The Optimum Design Speed (where tire wear is zero): $$\mathbf{v_{ ext{opt}} = \sqrt{rg an heta}}$$

Case 2: Rough Banked Road with Friction ($\mu_s$):

When the vehicle's speed differs from $v_{ ext{opt}}$, static friction acts along the banked incline to prevent slipping:

• Maximum Safe Speed ($v_{\max}$) without skidding outward (friction acts down the incline): $$\mathbf{v_{\max} = \sqrt{rg\left(rac{\mu_s + an heta}{1 - \mu_s an heta} ight)}}$$
• Minimum Safe Speed ($v_{\min}$) without sliding down (friction acts up the incline, if $ an heta > \mu_s$): $$\mathbf{v_{\min} = \sqrt{rg\left(rac{ an heta - \mu_s}{1 + \mu_s an heta} ight)}}$$

When a cyclist navigates a curve of radius $r$ at speed $v$, they lean inward toward the center of the curve by an angle $ heta$ with the vertical. The ground reaction force $R$ passes through the center of gravity, providing both normal support and centripetal force: