Unit 3: Laws of Motion

Laws of Motion is one of the highest-weight units in NEET Physics — expect 3–4 MCQs per year, spanning Newton's laws, momentum conservation, friction, and circular motion. The questions are typically free-body-diagram numericals, conceptual tests on the three laws, friction-on-incline numericals, and banking/conical-pendulum formula recall.

The unit rewards a clean habit of drawing FBDs and writing $\sum F = ma$ for each body and each direction.

Concept Map

• Inertia and Newton's three laws
• Linear momentum, impulse, conservation
• Equilibrium (concurrent forces)
• Friction (static, kinetic, rolling; angle of friction and repose)
• Inclined plane (with/without friction)
• Circular motion dynamics
  • Centripetal force
  • Banking of roads
  • Conical pendulum
  • Vertical circle
• Pseudo forces (non-inertial frames; qualitative)

Topic 1: Newton's Laws

Sub-topic A: First Law (Law of Inertia)

A body continues in its state of rest or uniform motion in a straight line unless acted upon by a net external force. This defines an inertial frame. Inertia comes in three flavours: of rest, of motion, of direction. Mass is the quantitative measure of inertia.

Sub-topic B: Second Law

The rate of change of linear momentum is proportional to the net force:

If mass is constant, $\vec F_\text{net} = m \vec a$.

For a variable-mass system (rocket), $\vec F_\text{net} = m\,\vec a + \vec v\,(dm/dt)$.

Sub-topic C: Third Law

For every action there is an equal and opposite reaction, acting on a different body. Action and reaction never cancel because they act on different bodies. Internal force pairs in a system sum to zero.

Topic 2: Momentum and Impulse

Sub-topic A: Linear Momentum

Sub-topic B: Impulse

Impulse $=$ change in momentum:

If force is constant in time, $\vec J = \vec F \cdot \Delta t$. Useful for short collisions where the average force can be estimated from $\langle F \rangle = \Delta p / \Delta t$.

The area under an $F$-$t$ graph is the impulse.

Sub-topic C: Conservation of Linear Momentum

If $\vec F_\text{ext} = 0$ for a system, $\vec p_\text{total} =$ constant. This is independent of whether internal forces are conservative.

Applications:

• Recoil: If a gun of mass $M$ fires a bullet of mass $m$ at speed $v$, gun recoils at $V = mv/M$.
• Rocket: Thrust $F = u\,(dm/dt)$ where $u$ is exhaust speed and $dm/dt$ is rate of mass ejection. Velocity gained, $v = u \ln(m_0/m)$ (Tsiolkovsky formula).

Topic 3: Equilibrium and Free-Body Diagrams

Sub-topic A: Equilibrium

A particle is in equilibrium if $\sum \vec F = 0$. For three concurrent forces in equilibrium, Lami's theorem holds:

where $\alpha, \beta, \gamma$ are the angles opposite to $F_1, F_2, F_3$.

Sub-topic B: Common FBD Components

Sub-topic C: Connected Bodies and Atwood Machine

Atwood machine (two masses $m_1 > m_2$ connected over smooth pulley):

Topic 4: Friction

Sub-topic A: Types

• Static friction $f_s$ adjusts to oppose tendency, up to $f_s^\text{max} = \mu_s N$.
• Kinetic friction $f_k = \mu_k N$, constant in magnitude during sliding (independent of speed to a good approximation).
• Rolling friction $f_r = \mu_r N$, with $\mu_r \ll \mu_k < \mu_s$.

Sub-topic B: Angle of Friction and Angle of Repose

If $\mu$ is the coefficient of friction, the angle of friction $\lambda$ is defined by $\tan\lambda = \mu$.

The angle of repose is the maximum angle of incline at which a body just starts sliding. It equals the angle of friction:

Sub-topic C: Friction on Horizontal Surface

To pull a block of mass $m$ at uniform velocity over a rough surface, the applied horizontal force is $F = \mu_k m g$. If pulled at an angle $\theta$ above horizontal:

The minimum force is at $\tan\theta = \mu_k$, giving $F_\text{min} = \mu_k m g/\sqrt{1 + \mu_k^2}$.

Sub-topic D: Friction on an Incline

For a block of mass $m$ on a rough incline of angle $\theta$:

• Required friction to prevent sliding down: $f = mg\sin\theta$. Possible if $\mu_s \ge \tan\theta$.
• Net acceleration down a slipping incline: $a = g(\sin\theta - \mu_k \cos\theta)$.
• Net acceleration up the incline (when applied force pushes upward): $a = -g(\sin\theta + \mu_k \cos\theta)$ during deceleration.

If $\theta < \theta_\text{repose}$, the block stays put.

Sub-topic E: Stopping Distance with Friction

A block of speed $v_0$ decelerated by friction on flat ground travels

Topic 5: Circular Motion Dynamics

Sub-topic A: Centripetal Acceleration and Force

For uniform circular motion of speed $v$ on a circle of radius $r$:

Centripetal force is toward the centre; it is not a new force — it is whatever real force (gravity, tension, friction, normal) provides this inward pull.

Sub-topic B: Banking of Roads

For a road banked at angle $\theta$ with no friction, the optimum speed is

With friction (coefficient $\mu$), maximum and minimum safe speeds are

If $\tan\theta \ge \mu$ no minimum exists (car will not slip inward).

For flat road (θ = 0): max safe speed $v_\text{max} = \sqrt{\mu r g}$.

Sub-topic C: Conical Pendulum

A bob of mass $m$ on string of length $\ell$ swings in a horizontal circle making angle $\theta$ with vertical. Then:

with $r = \ell \sin\theta$. Solving:

Sub-topic D: Vertical Circle

A body tied to a string of length $\ell$ swung in a vertical circle:

• At the top, minimum speed satisfies $m v_\text{top}^2/\ell = mg$, so $v_\text{top,min} = \sqrt{g\ell}$, and tension there can be zero.
• At the bottom, using energy conservation from top: $v_\text{bot}^2 = v_\text{top}^2 + 4g\ell$. With minimum top speed: $v_\text{bot,min} = \sqrt{5g\ell}$.
• Tension at the bottom (minimum case): $T_\text{bot} = 6 mg$.
• Tension at the top (minimum case): $T_\text{top} = 0$.
• General relation: $T_\text{bot} - T_\text{top} = 6 mg$ (always, regardless of speed).

For a bead on a rigid rod the minimum speed at the top is zero (rod can push, string cannot).

Sub-topic E: Death-Well (Wall of Death)

A motorcyclist on the vertical wall of a cylindrical well of radius $r$. Friction supports weight:

Topic 6: Pseudo Forces (Non-Inertial Frames)

In an accelerating frame with acceleration $\vec A$, add a pseudo force $-m\vec A$ to every body. Examples:

• Lift accelerating up at $a$: apparent weight $= m(g + a)$.
• Lift accelerating down at $a$: apparent weight $= m(g - a)$.
• Free-fall lift ($a = g$): apparent weight = 0 (weightlessness).

NEET Pattern MCQ Tips

• FBD numerical: typically Atwood, two-block system on smooth/rough surface, block on wedge.
• Friction recall: $\mu = \tan\theta_\text{repose}$; minimum force formula.
• Banking: optimum speed, max/min safe speed.
• Vertical circle: $v_\text{min}$ at top = $\sqrt{g\ell}$, tension difference = $6 mg$.
• Conical pendulum: $T = 2\pi\sqrt{\ell\cos\theta/g}$.
• Assertion-reason on action-reaction: pair acts on different bodies.
• Apparent weight in a lift.

Common Confusions and Traps

• Centripetal force is not a separate physical force; it's the net inward force.
• Friction is self-adjusting up to $\mu_s N$; it equals applied force only if the body is not sliding.
• Static friction can be less than $\mu_s N$ — it equals it only at the verge of slipping.
• Action and reaction act on different bodies, so they do not cancel.
• Normal force is not always $mg$ (changes on incline, in lift, in banked road).
• For a body on a smooth incline, only the component $mg\sin\theta$ is unbalanced.
• A car on a flat road takes turns by friction, not by banking.

Quick Revision Card

• $\vec F = d\vec p/dt$, impulse $= \Delta \vec p = \int F\,dt$.
• Atwood: $a = (m_1 - m_2)g/(m_1 + m_2)$, $T = 2 m_1 m_2 g/(m_1 + m_2)$.
• $\mu = \tan\theta_\text{repose}$; min pull angle $\tan\theta = \mu$.
• Inclined plane slipping: $a = g(\sin\theta - \mu\cos\theta)$.
• Banked road no friction: $v_0 = \sqrt{rg\tan\theta}$.
• Banked road with $\mu$: $v_\text{max} = \sqrt{rg(\tan\theta + \mu)/(1 - \mu\tan\theta)}$.
• Conical pendulum period: $T = 2\pi\sqrt{\ell\cos\theta/g}$.
• Vertical circle (string): $v_\text{top,min} = \sqrt{g\ell}$, $v_\text{bot,min} = \sqrt{5g\ell}$, $T_\text{bot} - T_\text{top} = 6mg$.
• Wall of death: $v_\text{min} = \sqrt{rg/\mu}$.
• Lift up at $a$: apparent weight = $m(g + a)$.

Worked NEET Examples

Example 1: Block on a horizontally accelerated wedge

A block of mass $m$ rests on a smooth wedge of angle $\theta$. The wedge accelerates horizontally at $a$ so the block stays at rest relative to wedge. Then in the wedge frame, pseudo force $-ma$ acts on the block. For block to be in equilibrium: $\tan\theta = a/g$, so $a = g\tan\theta$.

Example 2: Tension in a string supporting a hanging mass in a lift

Mass $m$ hangs from spring scale in a lift. Spring tension:

• Lift at rest or constant $v$: $T = mg$.
• Lift accelerating up at $a$: $T = m(g + a)$.
• Lift accelerating down at $a$: $T = m(g - a)$.
• Lift in free fall ($a = g$): $T = 0$.

Example 3: Block on a block

Block $m_1$ on block $m_2$, both on a smooth floor. Force $F$ applied horizontally to $m_2$. Both blocks share acceleration $a = F/(m_1 + m_2)$. Friction between blocks provides $m_1 a$. For $m_1$ to stay on $m_2$ without slipping: $\mu_s m_1 g \ge m_1 a$, so $F \le \mu_s (m_1 + m_2) g$.

Example 4: Inclined plane with applied horizontal force

Block of mass $m$ on a rough incline ($\theta, \mu$). Horizontal force $F$ applied. For block to be on the verge of moving up the incline:

Example 5: Minimum coefficient of friction for car turning

A car of mass $M$ travels at speed $v$ around a flat curve of radius $r$. Required centripetal force: $Mv^2/r$. This must be supplied by friction $\mu_s Mg$. So

For $v = 20$ m/s, $r = 100$ m, $g = 10$: $\mu_\text{min} = 400/(1000) = 0.4$.

Derivations Summary

Banking with Friction

On a banked road of angle $\theta$ with friction coefficient $\mu$:

Resolving along the horizontal (centripetal): $N\sin\theta + f\cos\theta = Mv^2/r$. Resolving along the vertical: $N\cos\theta - f\sin\theta = Mg$ where $f = \mu N$ at max speed.

Dividing: $\tan\theta + \mu)/(1 - \mu\tan\theta) = v^2/(rg)$, so

Conical Pendulum Period

Bob of mass $m$ moves in horizontal circle of radius $r = L\sin\theta$. Tension $T$ acts along string.

Vertical: $T\cos\theta = mg$. Horizontal: $T\sin\theta = m\omega^2 r$.

Dividing: $\tan\theta = \omega^2 r/g = \omega^2 L\sin\theta/g$, so

Vertical Circle Minimum Speed

At the top, minimum speed is when tension = 0, gravity alone provides centripetal force:

By energy conservation from top to bottom:

so $v_\text{bot}^2 = v_\text{top}^2 + 4gL = 5gL$. Hence $v_\text{bot,min} = \sqrt{5gL}$.

Tension at bottom: $T_\text{bot} - mg = m v_\text{bot}^2/L$, so $T_\text{bot} = m(g + 5g) = 6mg$.

Friction-Powered Acceleration

A block of mass $m$ on a rough surface ($\mu$) is pushed by force $F$ at angle $\theta$ above horizontal. Normal force: $N = mg - F\sin\theta$. Friction: $\mu N = \mu(mg - F\sin\theta)$. Net horizontal: $F\cos\theta - \mu(mg - F\sin\theta) = ma$.

For minimum $F$ to just start moving (a = 0): $F(\cos\theta + \mu\sin\theta) = \mu m g$, so

Differentiating with respect to $\theta$ and setting to zero gives $\tan\theta = \mu$, so $\theta_\text{opt} = \tan^{-1}\mu$ and

Special Force-Diagram Topics

Pulley with Mass

For a pulley of moment of inertia $I$ and radius $R$, with $m_1, m_2$ hanging on either side, the linear acceleration is

For a massless pulley, $I = 0$, recovering Atwood result.

Variable-Mass Systems

A rocket loses mass at rate $dm/dt$ (negative). The thrust force is

where $u$ is the exhaust speed relative to the rocket. Equation of motion:

leading (when $g$ ignored) to Tsiolkovsky: $v_f - v_i = u\ln(m_i/m_f)$.

Pseudo-Force Examples

A bob hangs from the ceiling of a vehicle. The vehicle accelerates horizontally at $a$. In the vehicle's frame, pseudo force $-ma$ acts on bob. Bob settles at angle $\theta = \tan^{-1}(a/g)$ behind the vertical.

In a uniformly rotating frame at angular velocity $\omega$, the pseudo forces are centrifugal ($m\omega^2 r$ outward) and Coriolis ($2m\vec v \times \vec\omega$). The Coriolis force is responsible for Earth's wind patterns (Northern Hemisphere: deflection to the right).

Formula Sheet