NEET Physics — Mechanics & Rotational Motion Formula Sheet

This comprehensive formula sheet is designed to be your ultimate companion for mastering the Mechanics and Rotational Motion sections of the NEET Physics syllabus. These topics are foundational, forming the bedrock of many other concepts in physics, and consistently carry significant weight in the examination. A thorough understanding and quick recall of these formulas are crucial for success.

NEET Physics: Mechanics & Rotational Motion Formula Sheet

Introduction to Mechanics & Rotational Motion

Mechanics is the branch of physics concerned with the motion of various physical objects and the forces that cause them. It is broadly divided into Statics (study of objects at rest) and Dynamics (study of objects in motion). Dynamics is further categorized into Kinematics (description of motion without considering its causes) and Kinetics (study of motion considering forces). Rotational Motion is a specialized area within mechanics, dealing with the movement of rigid bodies around an axis. It closely parallels linear motion, with analogous quantities and principles.

Exam Weightage Analysis

Mechanics and Rotational Motion collectively constitute a significant portion of the NEET Physics paper. Historically, these chapters contribute roughly 25-30% of the total physics questions, making them high-yield topics.

• Kinematics (1D & 2D): 2-3 questions
• Newton's Laws of Motion (NLM) & Friction: 3-4 questions
• Work, Energy & Power: 2-3 questions
• Centre of Mass & Collisions: 1-2 questions
• Rotational Motion: 2-3 questions
• Gravitation: 2-3 questions
• Properties of Matter (Solids & Fluids): 3-4 questions

Mastery of these concepts is therefore indispensable for achieving a high score in NEET Physics.

I. Kinematics

Kinematics describes motion without considering the forces causing it.

A. One-Dimensional Motion (Motion in a Straight Line)

• Average Velocity: $v_{avg} = \frac{\text{Total Displacement}}{\text{Total Time Taken}} = \frac{\Delta x}{\Delta t}$

• Average Speed: $Speed_{avg} = \frac{\text{Total Distance}}{\text{Total Time Taken}}$

• Instantaneous Velocity: $v = \frac{dx}{dt}$

• Instantaneous Acceleration: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ Also, $a = v \frac{dv}{dx}$ (useful when 'a' is a function of 'x')

• Equations of Motion (for constant acceleration $a$):

  1. $v = u + at$
  2. $s = ut + \frac{1}{2}at^2$
  3. $v^2 = u^2 + 2as$
  4. $s_n = u + \frac{a}{2}(2n - 1)$ (Displacement in the $n$-th second)

  Where:

  • $u$: initial velocity
  • $v$: final velocity
  • $s$: displacement
  • $t$: time
  • $a$: constant acceleration

• Free Fall (Motion under gravity):

  • When an object is thrown upwards or falls downwards under gravity, $a = g$ (downwards, approximately $9.8 \text{ m/s}^2$ or $10 \text{ m/s}^2$).
  • Take the direction of motion as positive or negative consistently. Generally, upward is positive, downward is negative.
  • At maximum height, $v = 0$.
  • Time of flight (when thrown up and returns to initial position): $T = \frac{2u}{g}$
  • Maximum height reached: $H_{max} = \frac{u^2}{2g}$

B. Two-Dimensional Motion (Projectile Motion)

• Projectile Motion: Motion where the object moves in a vertical plane under gravity, only considering initial velocity and gravity.

  • Horizontal acceleration $a_x = 0$.
  • Vertical acceleration $a_y = -g$ (taking upward as positive).

• Initial Velocity: $u$ at an angle $\theta$ with the horizontal.

  • Horizontal component: $u_x = u \cos\theta$
  • Vertical component: $u_y = u \sin\theta$

• Motion along x-axis:

  • $v_x = u_x = u \cos\theta$ (constant)
  • $x = u_x t = (u \cos\theta) t$

• Motion along y-axis:

  • $v_y = u_y - gt = u \sin\theta - gt$
  • $y = u_y t - \frac{1}{2}gt^2 = (u \sin\theta) t - \frac{1}{2}gt^2$

• Resultant Velocity at any time $t$:

  • $V = \sqrt{v_x^2 + v_y^2}$
  • Angle $\alpha$ with horizontal: $\tan\alpha = \frac{v_y}{v_x}$

• Time of Flight ($T$): The total time the projectile is in the air.

  • $T = \frac{2u \sin\theta}{g}$

• Maximum Height ($H_{max}$): The greatest vertical distance achieved.

  • $H_{max} = \frac{u^2 \sin^2\theta}{2g}$

• Horizontal Range ($R$): The total horizontal distance covered.

  • $R = \frac{u^2 \sin(2\theta)}{g}$
  • Maximum range occurs at $\theta = 45^\circ$, $R_{max} = \frac{u^2}{g}$.
  • Ranges are equal for complementary angles: $R(\theta) = R(90^\circ - \theta)$.

• Equation of Trajectory (path of the projectile):

  • $y = x \tan\theta - \frac{g x^2}{2 u^2 \cos^2\theta}$

C. Relative Motion

• Relative Velocity:

  • Velocity of object A with respect to object B: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$
  • Velocity of object B with respect to object A: $\vec{v}_{BA} = \vec{v}_B - \vec{v}A = -\vec{v}{AB}$

• Relative Position: $\vec{r}_{AB} = \vec{r}_A - \vec{r}_B$

• Relative Acceleration: $\vec{a}_{AB} = \vec{a}_A - \vec{a}_B$

• Special Cases:

  • Rain-Man Problems: $\vec{v}_{R,M} = \vec{v}_R - \vec{v}M$. The man aims to hold the umbrella relative to the direction of $\vec{v}{R,M}$.
  • River Boat Problems:
    • Velocity of boat in still water: $\vec{v}_b$
    • Velocity of river water: $\vec{v}_w$
    • Velocity of boat relative to ground: $\vec{v}_{bg} = \vec{v}_b + \vec{v}_w$
    • Shortest time to cross: Boat rows perpendicular to river flow.
      • Time $t = \frac{W}{v_b}$ (where $W$ is river width)
      • Drift $x = v_w t = \frac{v_w W}{v_b}$
    • Shortest path (no drift): Boat rows at an angle such that $\vec{v}_{bg}$ is perpendicular to river flow.
      • $\sin\theta = \frac{v_w}{v_b}$
      • Resultant speed $v_{bg} = \sqrt{v_b^2 - v_w^2}$
      • Time $t = \frac{W}{\sqrt{v_b^2 - v_w^2}}$
      • Condition: $v_b > v_w$ for shortest path across.

II. Newton's Laws of Motion (NLM) & Friction

A. Newton's Laws

• Newton's First Law (Law of Inertia): An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
• Newton's Second Law: The rate of change of momentum of an object is directly proportional to the applied unbalanced force in the direction of the force.
  • $\vec{F}_{net} = \frac{d\vec{p}}{dt}$
  • If mass ($m$) is constant: $\vec{F}_{net} = m\vec{a}$
  • Unit of Force: Newton (N) = kg m/s$^2$
• Newton's Third Law: For every action, there is an equal and opposite reaction.
  • $\vec{F}{AB} = -\vec{F}{BA}$
  • Action and reaction forces act on different bodies. They are always of the same magnitude and opposite direction.

B. Impulse and Momentum

• Linear Momentum: $\vec{p} = m\vec{v}$

• Impulse ($J$): Change in momentum.

  • $J = \Delta p = p_f - p_i = m(v_f - v_i)$
  • Also, $J = \int_{t_i}^{t_f} \vec{F} dt$
  • For constant force over time $\Delta t$: $J = \vec{F} \Delta t$
  • Unit of Impulse: N s or kg m/s

• Conservation of Linear Momentum: In the absence of external forces, the total linear momentum of a system remains constant.

  • $\Sigma \vec{p}{initial} = \Sigma \vec{p}{final}$
  • Especially useful in collision problems.

C. Connected Bodies & Pulleys

• Atwood Machine: Two masses $m_1$ and $m_2$ connected by a light string over a frictionless pulley.

  • Acceleration ($a$): $a = \frac{(m_1 - m_2)g}{m_1 + m_2}$ (assuming $m_1 > m_2$)
  • Tension ($T$): $T = \frac{2m_1m_2g}{m_1 + m_2}$

• Mass on a horizontal surface connected to a hanging mass:

  • Assuming frictionless surface:
    • Acceleration ($a$): $a = \frac{m_2 g}{m_1 + m_2}$
    • Tension ($T$): $T = \frac{m_1 m_2 g}{m_1 + m_2}$
  • Considering friction on the horizontal surface (coefficient $\mu_k$):
    • Acceleration ($a$): $a = \frac{m_2 g - \mu_k m_1 g}{m_1 + m_2}$
    • Tension ($T$): $T = \frac{m_1 m_2 g (1+\mu_k)}{m_1+m_2}$ (This needs re-derivation. $T = m_1 a + \mu_k m_1 g = m_1 (\frac{m_2 g - \mu_k m_1 g}{m_1 + m_2}) + \mu_k m_1 g = \frac{m_1 m_2 g - \mu_k m_1^2 g + \mu_k m_1^2 g + \mu_k m_1 m_2 g}{m_1 + m_2} = \frac{m_1 m_2 g(1 + \mu_k)}{m_1+m_2}$)
    • A simpler expression for tension is $T = m_1 a + \mu_k N_1 = m_1 a + \mu_k m_1 g$. Use this to derive $T$.
    • $m_2g - T = m_2 a$
    • $T - \mu_k m_1 g = m_1 a$
    • Adding both equations: $m_2g - \mu_k m_1 g = (m_1+m_2)a \implies a = \frac{(m_2 - \mu_k m_1)g}{m_1+m_2}$
    • Substitute $a$ into $T - \mu_k m_1 g = m_1 a$: $T = \mu_k m_1 g + m_1 \frac{(m_2 - \mu_k m_1)g}{m_1+m_2} = \frac{\mu_k m_1(m_1+m_2)g + m_1(m_2 - \mu_k m_1)g}{m_1+m_2} = \frac{\mu_k m_1^2 g + \mu_k m_1 m_2 g + m_1 m_2 g - \mu_k m_1^2 g}{m_1+m_2} = \frac{m_1 m_2 g (1+\mu_k)}{m_1+m_2}$

D. Friction

• Static Friction ($f_s$): Force that opposes the tendency of relative motion between surfaces in contact when they are at rest.

  • $f_s \le \mu_s N$
  • $\mu_s$: coefficient of static friction
  • $N$: normal force
  • $f_s^{max} = \mu_s N$ (limiting static friction)

• Kinetic Friction ($f_k$): Force that opposes relative motion between surfaces in contact when they are already sliding.

  • $f_k = \mu_k N$
  • $\mu_k$: coefficient of kinetic friction
  • Generally, $\mu_k < \mu_s$.

• Angle of friction ($\phi_s$): Angle between the resultant of normal force and limiting friction, and the normal force.

  • $\tan\phi_s = \mu_s$

• Angle of Repose ($\theta_r$): The minimum angle of inclination of a plane with the horizontal such that a body placed on it just begins to slide down.

  • $\tan\theta_r = \mu_s$

III. Work, Energy & Power

A. Work

• Work Done by a Constant Force:
  • $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$
  • $\theta$: angle between force and displacement vectors.
  • Unit of Work: Joule (J) = N m
• Work Done by a Variable Force:
  • $W = \int \vec{F} \cdot d\vec{r}$
  • For one-dimensional motion: $W = \int_{x_i}^{x_f} F(x) dx$
• Work-Energy Theorem: The net work done on an object equals its change in kinetic energy.
  • $W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

B. Energy

• Kinetic Energy (KE): Energy due to motion.

  • $K = \frac{1}{2}mv^2$

• Potential Energy (PE): Energy due to position or configuration.

  • Gravitational Potential Energy: $U_g = mgh$ (for uniform gravity)
  • Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ (where $k$ is spring constant, $x$ is compression/extension)

• Conservation of Mechanical Energy: If only conservative forces (gravity, spring force) are doing work, total mechanical energy ($E = K + U$) is conserved.

  • $K_i + U_i = K_f + U_f$

• Conservative vs. Non-Conservative Forces:

  • Conservative: Work done is independent of the path taken; (e.g., gravity, elastic force). Can be expressed as $F = -\frac{dU}{dx}$.
  • Non-Conservative: Work done depends on the path taken; (e.g., friction, air resistance). Dissipates energy.
  • When non-conservative forces are present: $W_{nc} = \Delta E = (K_f + U_f) - (K_i + U_i)$

C. Power

• Average Power: $P_{avg} = \frac{\Delta W}{\Delta t}$
• Instantaneous Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$
• Unit of Power: Watt (W) = J/s
• Relation between Power and Force: $P = Fv \cos\theta$ (if constant force and velocity direction)

IV. Centre of Mass (COM) & Collisions

A. Centre of Mass

• For a system of $n$ discrete particles:

  • Position vector of COM: $\vec{r}_{com} = \frac{\Sigma m_i \vec{r}_i}{\Sigma m_i}$
  • x-coordinate of COM: $x_{com} = \frac{\Sigma m_i x_i}{\Sigma m_i}$
  • y-coordinate of COM: $y_{com} = \frac{\Sigma m_i y_i}{\Sigma m_i}$
  • z-coordinate of COM: $z_{com} = \frac{\Sigma m_i z_i}{\Sigma m_i}$

• For a continuous body:

  • Position vector of COM: $\vec{r}_{com} = \frac{\int \vec{r} dm}{\int dm} = \frac{1}{M} \int \vec{r} dm$
  • $dm = \lambda dx$ (for line density $\lambda$)
  • $dm = \sigma dA$ (for surface density $\sigma$)
  • $dm = \rho dV$ (for volume density $\rho$)

• Velocity of COM: $\vec{v}_{com} = \frac{\Sigma m_i \vec{v}i}{\Sigma m_i} = \frac{d\vec{r}{com}}{dt}$

• Acceleration of COM: $\vec{a}_{com} = \frac{\Sigma m_i \vec{a}i}{\Sigma m_i} = \frac{d\vec{v}{com}}{dt}$

• Newton's Second Law for a system of particles: $\vec{F}{ext} = M\vec{a}{com}$ (where $M = \Sigma m_i$ is total mass).

  • If $\vec{F}{ext} = 0$, then $\vec{a}{com} = 0$, and $\vec{v}{com} = \text{constant}$. This means the total momentum of the system ($\vec{P}{total} = M\vec{v}_{com}$) is conserved.

B. Collisions

• Collision: An isolated event in which two or more bodies exert forces on each other for a relatively short time.

• Key Principle: Linear momentum is always conserved in any type of collision if no external forces act on the system.

  • $\Sigma \vec{p}{initial} = \Sigma \vec{p}{final}$

• Types of Collisions:

  • Elastic Collision: Both momentum and kinetic energy are conserved.
    • $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$
    • $\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$
    • Relative speed of approach = Relative speed of separation: $(u_1 - u_2) = -(v_1 - v_2)$
    • Coefficient of Restitution ($e$): $e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} = \frac{-(v_1 - v_2)}{(u_1 - u_2)}$
    • For elastic collision, $e = 1$.
    • Velocities after 1D elastic collision:
      • $v_1 = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)u_1 + \left(\frac{2m_2}{m_1 + m_2}\right)u_2$
      • $v_2 = \left(\frac{2m_1}{m_1 + m_2}\right)u_1 + \left(\frac{m_2 - m_1}{m_1 + m_2}\right)u_2$
  • Inelastic Collision: Momentum is conserved, but kinetic energy is not conserved (some KE is lost as heat, sound, deformation).
    • For inelastic collision, $0 < e < 1$.
  • Perfectly Inelastic Collision: Bodies stick together after collision ($v_1 = v_2 = v$). Kinetic energy loss is maximum.
    • For perfectly inelastic collision, $e = 0$.
    • $m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$
    • Final velocity: $v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$

V. Rotational Motion

A. Rotational Kinematics

• Angular Position: $\theta$ (in radians)

• Angular Displacement: $\Delta\theta$

• Average Angular Velocity: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$

• Instantaneous Angular Velocity: $\omega = \frac{d\theta}{dt}$

• Average Angular Acceleration: $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$

• Instantaneous Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$

• Equations of Rotational Motion (for constant angular acceleration $\alpha$):

  1. $\omega = \omega_0 + \alpha t$
  2. $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
  3. $\omega^2 = \omega_0^2 + 2\alpha\theta$
  4. $\theta_n = \omega_0 + \frac{\alpha}{2}(2n - 1)$

• Relationship between Linear & Angular Variables (for a point on a rotating rigid body):

  • $\vec{v} = \vec{\omega} \times \vec{r}$ (velocity vector)
  • Tangential Acceleration: $a_t = r\alpha$ (changes speed)
  • Centripetal (Radial) Acceleration: $a_c = \frac{v^2}{r} = r\omega^2$ (changes direction, always towards center)
  • Total Acceleration: $\vec{a} = \vec{a}_t + \vec{a}_c$. Magnitude: $a = \sqrt{a_t^2 + a_c^2}$

• Period ($T$) and Frequency ($f$):

  • $T = \frac{2\pi}{\omega}$
  • $f = \frac{1}{T} = \frac{\omega}{2\pi}$

B. Moment of Inertia (I)

• Rotational Inertia: Analogous to mass in linear motion. Measures resistance to change in rotational motion.

  • For a system of discrete particles: $I = \Sigma m_i r_i^2$
  • For a continuous body: $I = \int r^2 dm$
  • Unit: kg m$^2$

• Theorems of Moment of Inertia:

  • Parallel Axis Theorem: $I = I_{cm} + Md^2$
    • $I_{cm}$: Moment of inertia about an axis passing through the center of mass.
    • $M$: Total mass of the body.
    • $d$: Perpendicular distance between the two parallel axes.
  • Perpendicular Axis Theorem (for planar bodies only): $I_z = I_x + I_y$
    • $I_x, I_y$: MOI about two perpendicular axes in the plane of the body.
    • $I_z$: MOI about an axis perpendicular to the plane, passing through their intersection.

• Common Moments of Inertia:

  Body	Axis	Moment of Inertia ($I$)
  Ring / Hollow Cylinder	Axis passing through center, perpendicular to plane (or along axis)	$MR^2$
  Disc / Solid Cylinder	Axis passing through center, perpendicular to plane (or along axis)	$\frac{1}{2}MR^2$
  Solid Sphere	Axis through center	$\frac{2}{5}MR^2$
  Hollow Sphere	Axis through center	$\frac{2}{3}MR^2$
  Rod (thin)	Axis through center, perpendicular to length	$\frac{1}{12}ML^2$
  Rod (thin)	Axis through one end, perpendicular to length	$\frac{1}{3}ML^2$

C. Torque ($\tau$)

• Rotational equivalent of force. Causes angular acceleration.

  • $\vec{\tau} = \vec{r} \times \vec{F}$
  • Magnitude: $\tau = rF \sin\theta = F \cdot r_{\perp}$ (where $r_{\perp}$ is perpendicular distance from the axis of rotation to the line of action of the force, called lever arm).
  • Unit: N m

• Newton's Second Law for Rotation:

  • $\vec{\tau}_{net} = I\vec{\alpha}$

D. Angular Momentum ($\vec{L}$)

• Rotational equivalent of linear momentum.

  • For a single particle: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$
  • Magnitude: $L = mvr \sin\theta = mv r_{\perp}$
  • For a rigid body rotating about a fixed axis: $\vec{L} = I\vec{\omega}$
  • Unit: kg m$^2$/s or J s

• Relation between Torque and Angular Momentum:

  • $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$

• Conservation of Angular Momentum: If the net external torque acting on a system is zero, its total angular momentum remains constant.

  • $\vec{L}{initial} = \vec{L}{final}$
  • $I_1 \omega_1 = I_2 \omega_2$

E. Work, Energy, and Power in Rotational Motion

• Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$

• Total Kinetic Energy (for rolling motion): $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2$

  • For pure rolling: $v_{cm} = R\omega$
  • $K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\left(\frac{v_{cm}}{R}\right)^2 = \frac{1}{2}Mv_{cm}^2 \left(1 + \frac{I_{cm}}{MR^2}\right)$
  • Radius of Gyration ($k$): $I = Mk^2$. So, $K_{total} = \frac{1}{2}Mv_{cm}^2 \left(1 + \frac{k^2}{R^2}\right)$
  • The term $\frac{k^2}{R^2}$ is a pure number, characteristic of the body's shape.
    • Ring: $\frac{k^2}{R^2} = 1$
    • Disc/Cylinder: $\frac{k^2}{R^2} = \frac{1}{2}$
    • Solid Sphere: $\frac{k^2}{R^2} = \frac{2}{5}$
    • Hollow Sphere: $\frac{k^2}{R^2} = \frac{2}{3}$

• Work Done by a Torque: $W = \int \tau d\theta$

  • For constant torque: $W = \tau \Delta\theta$

• Power in Rotational Motion: $P = \tau\omega$

F. Rolling Motion

• Pure Rolling: No relative slipping between the point of contact of the body and the surface.
  • Velocity of point of contact = $0$
  • Velocity for a point at distance $r'$ from COM: $v_P = \sqrt{v_{cm}^2 + (r'\omega)^2 + 2v_{cm}(r'\omega) \cos\phi}$
  • $v_{cm} = R\omega$ (for pure rolling)
• Acceleration of COM for a body rolling down an incline:
  • $a_{cm} = \frac{g \sin\theta}{1 + I_{cm}/MR^2} = \frac{g \sin\theta}{1 + k^2/R^2}$
• Friction force for rolling down an incline:
  • $f_s = \frac{Mg \sin\theta}{1 + MR^2/I_{cm}} = \frac{Mg \sin\theta}{1 + R^2/k^2}$
  • Rolling without slipping requires $f_s \le \mu_s N = \mu_s Mg \cos\theta$.
  • Thus, $\frac{g \sin\theta}{1 + R^2/k^2} \le \mu_s g \cos\theta \implies \tan\theta \le \mu_s \left(1 + \frac{R^2}{k^2}\right)$ is the condition for pure rolling.

VI. Gravitation

A. Newton's Law of Universal Gravitation

• Every particle attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
  • $F = G \frac{m_1 m_2}{r^2}$
  • $G$: Universal Gravitational Constant = $6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$

B. Acceleration Due to Gravity ($g$)

• On Earth's surface: $g = \frac{GM_E}{R_E^2}$, approximately $9.8 \text{ m/s}^2$
• Variation of $g$ with Altitude ($h$):
  • For $h \ll R_E$: $g' = g \left(1 - \frac{2h}{R_E}\right)$
  • For any $h$: $g' = \frac{GM_E}{(R_E + h)^2}$
• Variation of $g$ with Depth ($d$):
  • $g' = g \left(1 - \frac{d}{R_E}\right)$
• Variation of $g$ with Latitude ($\lambda$):
  • $g'(\lambda) = g - R_E \omega^2 \cos^2\lambda$
    • At poles ($\lambda = 90^\circ$): $g' = g$
    • At equator ($\lambda = 0^\circ$): $g' = g - R_E \omega^2$

C. Gravitational Potential Energy ($U$)

• $U = -\frac{GMm}{r}$ (measured from infinity)
• For an object at height $h$ from Earth's surface: $\Delta U = mgH_{effective}$ or $U = -\frac{GM_E m}{R_E + h}$

D. Gravitational Potential ($V$)

• Potential energy per unit mass.
  • $V = \frac{U}{m} = -\frac{GM}{r}$
• Relation to Field: $\vec{E} = -\nabla V = -\left(\hat{i}\frac{\partial V}{\partial x} + \hat{j}\frac{\partial V}{\partial y} + \hat{k}\frac{\partial V}{\partial z}\right)$
  • $E = -\frac{dV}{dr}$ (for radial field)

E. Escape Velocity ($v_e$)

• Minimum velocity required for an object to escape Earth's gravitational field.
  • $v_e = \sqrt{\frac{2GM_E}{R_E}} = \sqrt{2gR_E}$
  • Approximately $11.2 \text{ km/s}$ for Earth.

F. Orbital Velocity ($v_o$)

• Velocity required for an object to orbit around a planet in a stable circular orbit.
  • $v_o = \sqrt{\frac{GM}{r}}$ (where $r$ is orbital radius)
  • For an orbit just above Earth's surface ($r \approx R_E$): $v_o = \sqrt{gR_E} \approx 7.9 \text{ km/s}$

G. Time Period of Satellite ($T$)

• $T = \frac{2\pi r}{v_o} = 2\pi \sqrt{\frac{r^3}{GM}}$
• Geostationary Satellite: $T = 24 \text{ hours}$, $h \approx 36000 \text{ km}$ from Earth's surface.

H. Kepler's Laws of Planetary Motion

1. Law of Orbits: All planets move in elliptical orbits with the Sun at one of the foci.
2. Law of Areas: The line joining any planet to the Sun sweeps out equal areas in equal intervals of time. (Consequence of angular momentum conservation: $\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$)
3. Law of Periods: The square of the time period of any planet is proportional to the cube of the semi-major axis of its orbit.
   • $T^2 \propto a^3$ or $T^2 = \frac{4\pi^2}{GM} a^3$ (for circular orbit, $a=r$)

I. Binding Energy

• Energy required to remove a body from its gravitational influence.
  • Binding Energy $= -(\text{Total Mechanical Energy})$
  • For a satellite: $E = K + U = \frac{1}{2}mv_o^2 - \frac{GMm}{r} = \frac{1}{2}m\left(\frac{GM}{r}\right) - \frac{GMm}{r} = -\frac{GMm}{2r}$
  • Binding Energy $= \frac{GMm}{2r}$

VII. Properties of Matter

A. Elasticity

• Stress ($\sigma$): Restoring force per unit area.

  • $\sigma = \frac{F}{A}$ (Unit: N/m$^2$ or Pascal)

• Strain ($\epsilon$): Change in dimension per original dimension. (Dimensionless)

  • Longitudinal strain: $\epsilon_L = \frac{\Delta L}{L}$
  • Volumetric strain: $\epsilon_V = \frac{\Delta V}{V}$
  • Shear strain: $\epsilon_S = \frac{\Delta x}{h} = \tan\theta \approx \theta$

• Hooke's Law: Within elastic limit, stress is proportional to strain.

  • $\sigma = Y \epsilon$ (where $Y$ is modulus of elasticity)

• Moduli of Elasticity:

  • Young's Modulus ($Y$): For longitudinal stress and strain (solids).
    • $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L}$
  • Bulk Modulus ($B$): For volumetric stress and strain (solids, liquids, gases). Reciprocal is compressibility.
    • $B = \frac{\text{Volume Stress}}{\text{Volume Strain}} = \frac{-\Delta P}{\Delta V/V}$
  • Shear Modulus / Modulus of Rigidity ($G$ or $\eta$): For tangential stress and shear strain (solids).
    • $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\theta}$

• Poisson's Ratio ($\nu$):

  • $\nu = -\frac{\text{Lateral strain}}{\text{Longitudinal strain}} = -\frac{\Delta D/D}{\Delta L/L}$ (for a cylinder, $D$ is diameter)
  • The negative sign indicates that if length increases, diameter decreases. Typical values $0.2-0.5$.

• Elastic Potential Energy Density: Energy stored per unit volume.

  • $U_D = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2}Y(\text{Strain})^2 = \frac{1}{2}\frac{(\text{Stress})^2}{Y}$

B. Fluid Mechanics

• Density ($\rho$): $\rho = \frac{mass}{volume}$ (Unit: kg/m$^3$)

• Relative Density (Specific Gravity): Ratio of density of substance to density of reference ($1000 \text{ kg/m}^3$ for water at $4^\circ\text{C}$). Dimensionless.

• Pressure ($P$): Force per unit area.

  • $P = \frac{F}{A}$ (Unit: N/m$^2$ or Pascal)
  • Gauge Pressure: $P_{gauge} = P - P_{atm}$
  • Absolute Pressure: $P_{abs} = P_{atm} + P_{gauge}$

• Pressure in a fluid column: $P = \rho gh$ (at depth $h$).

• Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

  • Hydraulic Lift: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

• Buoyancy and Archimedes' Principle: Buoyant force $(F_B)$ on a submerged/partially submerged object is equal to the weight of the fluid displaced.

  • $F_B = V_{displaced} \rho_{fluid} g$
  • Apparent weight = Actual weight - Buoyant force $= W - F_B$
  • A floating object displaces its own weight of fluid.

• Equation of Continuity (for incompressible, non-viscous fluid, steady flow):

  • Product of area and velocity is constant: $A_1 v_1 = A_2 v_2$ (Mass flow rate is constant)

• Bernoulli's Equation (for incompressible, non-viscous fluid, steady flow along a streamline):

  • $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
  • States that sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant.
  • Torricelli's Law (Efflux velocity): $v = \sqrt{2gh}$ (velocity of efflux from an orifice at depth $h$ below free surface).

• Viscosity: Internal friction in fluids.

  • Viscous Force: $F = -\eta A \frac{dv}{dz}$ (Newton's Law of Viscosity)
    • $\eta$: coefficient of viscosity (Unit: Pa s or Poiseuille or Poise - $1 \text{ Pa s} = 10 \text{ Poise}$)
    • $\frac{dv}{dz}$: velocity gradient
  • Stokes' Law: Viscous drag force on a spherical body of radius $r$ moving with velocity $v$ in a fluid of viscosity $\eta$.
    • $F_D = 6\pi\eta rv$
  • Terminal Velocity ($v_T$): Constant velocity attained when viscous drag = buoyant force + weight.
    • $v_T = \frac{2r^2(\rho_{body} - \rho_{fluid})g}{9\eta}$

• Surface Tension ($\gamma$ or $S$ or $T_s$): Force per unit length acting tangentially to the free surface of a liquid.

  • $\gamma = \frac{F}{L}$ (Unit: N/m or J/m$^2$)

• Surface Energy: Work done to increase surface area by unit amount.

  • $W = \gamma \Delta A$

• Excess Pressure: Pressure difference across a curved liquid surface.

  • Liquid drop: $\Delta P = \frac{2\gamma}{R}$
  • Soap bubble: $\Delta P = \frac{4\gamma}{R}$

• Capillarity: Rise or fall of liquid in a capillary tube.

  • Height of rise ($h$): $h = \frac{2\gamma \cos\theta}{\rho gR}$
    • $\theta$: angle of contact
    • $R$: radius of capillary tube

Worked Examples

Example 1: Projectile Motion

A projectile is fired with an initial velocity of $20 \text{ m/s}$ at an angle of $30^\circ$ above the horizontal. Calculate its maximum height and horizontal range. (Take $g = 10 \text{ m/s}^2$)

Solution: Given: $u = 20 \text{ m/s}$, $\theta = 30^\circ$, $g = 10 \text{ m/s}^2$.

• Maximum Height ($H_{max}$): $$H_{max} = \frac{u^2 \sin^2\theta}{2g}$$ $$H_{max} = \frac{(20)^2 (\sin 30^\circ)^2}{2 \times 10}$$ $$H_{max} = \frac{400 \times (0.5)^2}{20}$$ $$H_{max} = \frac{400 \times 0.25}{20} = \frac{100}{20} = 5 \text{ m}$$

• Horizontal Range ($R$): $$R = \frac{u^2 \sin(2\theta)}{g}$$ $$R = \frac{(20)^2 \sin(2 \times 30^\circ)}{10}$$ $$R = \frac{400 \sin(60^\circ)}{10}$$ $$R = \frac{400 \times \frac{\sqrt{3}}{2}}{10} = \frac{200\sqrt{3}}{10} = 20\sqrt{3} \text{ m}$$ (Approx. $20 \times 1.732 = 34.64 \text{ m}$)

Example 2: Blocks on a Frictionless Surface

Two blocks of mass $m_1 = 2 \text{ kg}$ and $m_2 = 3 \text{ kg}$ are placed in contact on a frictionless horizontal surface. A horizontal force $F = 10 \text{ N}$ is applied to $m_1$. a) Find the acceleration of the system. b) Find the contact force between the two blocks.

Solution: Given: $m_1 = 2 \text{ kg}$, $m_2 = 3 \text{ kg}$, $F = 10 \text{ N}$. The surface is frictionless.

a) Acceleration of the system: Treat both blocks as a single system. The total mass is $M = m_1 + m_2 = 2 + 3 = 5 \text{ kg}$. The applied force $F$ acts on this combined system. Using Newton's Second Law ($\vec{F}_{net} = M\vec{a}$): $$F = (m_1 + m_2)a$$ $$10 = (2 + 3)a$$ $$10 = 5a$$ $$a = \frac{10}{5} = 2 \text{ m/s}^2$$ The acceleration of both blocks is $2 \text{ m/s}^2$.

b) Contact force between the blocks: Let $N_{12}$ be the contact force exerted by $m_1$ on $m_2$. By Newton's Third Law, the force exerted by $m_2$ on $m_1$ is $N_{21} = -N_{12}$. Consider only block $m_2$. The only horizontal force acting on $m_2$ is $N_{12}$ from $m_1$. Applying Newton's Second Law to $m_2$: $$N_{12} = m_2 a$$ $$N_{12} = 3 \text{ kg} \times 2 \text{ m/s}^2$$ $$N_{12} = 6 \text{ N}$$ Alternatively, consider block $m_1$. The forces on $m_1$ are $F$ (applied force) and $N_{21}$ (contact force from $m_2$). $$F - N_{21} = m_1 a$$ $$10 - N_{21} = 2 \text{ kg} \times 2 \text{ m/s}^2$$ $$10 - N_{21} = 4$$ $$N_{21} = 10 - 4 = 6 \text{ N}$$ The magnitude of the contact force is $6 \text{ N}$.

Example 3: Work-Energy Theorem

A block of mass $5 \text{ kg}$ is initially at rest on a rough horizontal surface. A horizontal force of $20 \text{ N}$ is applied to the block. If the coefficient of kinetic friction is $0.2$, calculate the work done by the applied force when the block moves $4 \text{ m}$. Also, find the final kinetic energy of the block. (Take $g = 10 \text{ m/s}^2$)

Solution: Given: $m = 5 \text{ kg}$, $u = 0$, $F_{app} = 20 \text{ N}$, $\mu_k = 0.2$, $s = 4 \text{ m}$.

• Work done by the applied force ($W_{app}$): Since the applied force is constant and in the direction of displacement: $$W_{app} = F_{app} \times s$$ $$W_{app} = 20 \text{ N} \times 4 \text{ m} = 80 \text{ J}$$

• Find the final kinetic energy of the block using the Work-Energy Theorem: First, calculate the net work done on the block. We need the kinetic friction force. Normal force $N = mg = 5 \text{ kg} \times 10 \text{ m/s}^2 = 50 \text{ N}$. Kinetic friction force $f_k = \mu_k N = 0.2 \times 50 \text{ N} = 10 \text{ N}$. The net force acting on the block is $F_{net} = F_{app} - f_k = 20 \text{ N} - 10 \text{ N} = 10 \text{ N}$. The net work done $W_{net} = F_{net} \times s = 10 \text{ N} \times 4 \text{ m} = 40 \text{ J}$.

  According to the Work-Energy Theorem: $$W_{net} = \Delta K = K_f - K_i$$ Since the block starts from rest, $K_i = 0$. $$K_f = W_{net}$$ $$K_f = 40 \text{ J}$$ The final kinetic energy of the block is $40 \text{ J}$.

Example 4: Conservation of Angular Momentum

A uniform disc of mass $M$ and radius $R$ is rotating about its axis with an angular speed $\omega_0$. A second identical disc is gently placed concentrically on the first disc. Both discs then rotate with a common angular speed $\omega$. Find $\omega$. Assume no external torque acts on the system.

Solution: Given: First disc: mass $M$, radius $R$, initial angular speed $\omega_0$. Second disc: identical (mass $M$, radius $R$).

• Moment of Inertia of a single disc: For a uniform disc rotating about its axis, $I = \frac{1}{2}MR^2$.

• Initial Angular Momentum ($L_i$): Initially, only the first disc is rotating. $$L_i = I_1 \omega_0 = \frac{1}{2}MR^2 \omega_0$$

• Final Angular Momentum ($L_f$): When the second disc is placed, they rotate together. The total moment of inertia ($I_f$) is the sum of the moments of inertia of the two discs. $$I_f = I_1 + I_2 = \frac{1}{2}MR^2 + \frac{1}{2}MR^2 = MR^2$$ Let the final common angular speed be $\omega$. $$L_f = I_f \omega = (MR^2)\omega$$

• Conservation of Angular Momentum: Since no external torque acts, angular momentum is conserved: $L_i = L_f$. $$\frac{1}{2}MR^2 \omega_0 = (MR^2)\omega$$ Cancel $MR^2$ from both sides: $$\frac{1}{2}\omega_0 = \omega$$ Therefore, the final angular speed of the combined system is $\omega = \frac{\omega_0}{2}$.

Example 5: Gravitation - Orbital Velocity

A satellite orbits Earth at a height of $3 \times 10^5 \text{ m}$ above the surface. Given Earth's radius $R_E = 6.4 \times 10^6 \text{ m}$ and acceleration due to gravity $g = 9.8 \text{ m/s}^2$, find the orbital speed of the satellite.

Solution: Given: Height $h = 3 \times 10^5 \text{ m}$, $R_E = 6.4 \times 10^6 \text{ m}$, $g = 9.8 \text{ m/s}^2$.

• Orbital Radius ($r$): The orbital radius is the distance from the center of the Earth to the satellite. $$r = R_E + h = 6.4 \times 10^6 \text{ m} + 3 \times 10^5 \text{ m} = 6.4 \times 10^6 \text{ m} + 0.3 \times 10^6 \text{ m} = 6.7 \times 10^6 \text{ m}$$

• Orbital Velocity ($v_o$): The formula for orbital velocity is $v_o = \sqrt{\frac{GM_E}{r}}$. We also know that $g = \frac{GM_E}{R_E^2}$, so $GM_E = gR_E^2$. Substitute $GM_E$ in the orbital velocity formula: $$v_o = \sqrt{\frac{gR_E^2}{r}}$$ $$v_o = \sqrt{\frac{9.8 \text{ m/s}^2 \times (6.4 \times 10^6 \text{ m})^2}{6.7 \times 10^6 \text{ m}}}$$ $$v_o = \sqrt{\frac{9.8 \times (6.4)^2 \times 10^{12}}{6.7 \times 10^6}}$$ $$v_o = \sqrt{\frac{9.8 \times 40.96 \times 10^6}{6.7}}$$ $$v_o = \sqrt{\frac{401.408 \times 10^6}{6.7}} \approx \sqrt{59.91 \times 10^6}$$ $$v_o \approx 7.74 \times 10^3 \text{ m/s} = 7.74 \text{ km/s}$$

Example 6: Fluid Mechanics - Capillary Action

Water rises to a height of $2 \text{ cm}$ in a capillary tube. If the radius of the tube is $0.5 \text{ mm}$, calculate the surface tension of water. Assume the angle of contact is $0^\circ$ and density of water is $1000 \text{ kg/m}^3$. ($g = 10 \text{ m/s}^2$)

Solution: Given: $h = 2 \text{ cm} = 0.02 \text{ m}$, $R = 0.5 \text{ mm} = 0.5 \times 10^{-3} \text{ m}$, $\theta = 0^\circ$, $\rho = 1000 \text{ kg/m}^3$, $g = 10 \text{ m/s}^2$.

• Formula for capillary rise: $$h = \frac{2\gamma \cos\theta}{\rho gR}$$ We need to find $\gamma$. Rearrange the formula: $$\gamma = \frac{h \rho gR}{2 \cos\theta}$$

• Substitute the values: $$\gamma = \frac{0.02 \text{ m} \times 1000 \text{ kg/m}^3 \times 10 \text{ m/s}^2 \times (0.5 \times 10^{-3}) \text{ m}}{2 \times \cos 0^\circ}$$ $$\gamma = \frac{0.02 \times 1000 \times 10 \times 0.5 \times 10^{-3}}{2 \times 1}$$ $$\gamma = \frac{200 \times 0.5 \times 10^{-3}}{2}$$ $$\gamma = \frac{100 \times 10^{-3}}{2}$$ $$\gamma = 50 \times 10^{-3} = 0.05 \text{ N/m}$$ The surface tension of water is $0.05 \text{ N/m}$.

Common Mistakes to Avoid

1. Sign Conventions: In Kinematics and NLM, consistently apply sign conventions for direction (e.g., upward positive, downward negative).
2. Vector vs. Scalar: Distinguish between vector quantities (displacement, velocity, force, momentum, angular momentum, torque) and scalar quantities (distance, speed, mass, energy, work, power, kinetic energy).
3. Units: Always check units and ensure consistency. Use SI units for calculations.
4. Friction: Remember that static friction adjusts its magnitude up to a maximum value, while kinetic friction is constant. Also, distinguish between $\mu_s$ and $\mu_k$.
5. Work by Gravity: Work done by gravity depends only on the vertical displacement, not the path.
6. Conservation Laws:
   • Momentum: Conserved always in collisions if no external force.
   • Kinetic Energy: Conserved only in elastic collisions.
   • Mechanical Energy: Conserved only if conservative forces do work.
   • Angular Momentum: Conserved if net external torque is zero.
7. COM vs. Centre of Gravity: For uniform gravity, COM and CG coincide. For large objects where gravity varies, they might differ slightly.
8. Rolling Motion: The point of contact in pure rolling has zero velocity. Do not confuse $v_{cm}$ with rotational speed $R\omega$ (it's $v_{cm}=R\omega$).
9. Parallel vs. Perpendicular Axis Theorem: Perpendicular axis theorem applies only to planar bodies.
10. Gravitation: Distinguish between gravitational force (between two masses) and acceleration due to gravity (field strength).
11. Fluids: Remember pressure in liquids depends only on depth for a static fluid, not on the shape of the container. Bernoulli's equation applies to ideal fluids under steady flow.

Memory Tricks / Mnemonics

• Kinematic Equations (Constant Acceleration):
  • SUVAT: $s$, $u$, $v$, $a$, $t$. Each equation misses one variable.
    1. $v = u + at$ (No $s$)
    2. $s = ut + \frac{1}{2}at^2$ (No $v$)
    3. $v^2 = u^2 + 2as$ (No $t$)
    4. $s = \frac{u+v}{2}t$ (No $a$)
• Vector Cross Product (Right Hand Rule): For $\vec{A} \times \vec{B}$, curl fingers from $\vec{A}$ to $\vec{B}$, thumb points in direction of $\vec{A} \times \vec{B}$.
• Moment of Inertia Factors ($\frac{k^2}{R^2}$):
  • Disc vs. Ring: Disc is solid, distributed closer to axis, so less inertia. (Disc $1/2$, Ring $1$). If you imagine rotating, it's easier to rotate a disc than a ring of same mass and radius about its axis.
  • Sphere: Solid sphere $2/5$, Hollow sphere $2/3$. Hollow has more mass at periphery, so more inertia. $2/3 > 2/5$.
• Bernoulli's Equation: "$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$" All terms are energy per unit volume (or pressure terms). "Pressure + Kinetic Energy/Volume + Potential Energy/Volume = Constant"
• Capillary Rise: "2T cos $\theta$ over $\rho g r$" - (2-times-gamma-cos-theta over rho-g-r) $$h = \frac{2\gamma \cos\theta}{\rho gR}$$
• Escape Velocity vs. Orbital Velocity: $v_e = \sqrt{2} v_o$. Escaping requires more energy/velocity than just orbiting.

Previous Year Question (PYQ) Patterns

• Kinematics: Direct application of equations of motion, especially for motion under gravity. Relative motion problems (rain-man, river-boat) are frequent. Projectile motion questions often involve finding range, max height, or striking points.
• NLM & Friction: Drawing Free Body Diagrams (FBDs) is crucial. Problems involving connected masses (pulleys, blocks) are very common. Friction questions often test limiting friction and conditions for motion/rest. Impulse-momentum theorem is also tested.
• Work, Energy & Power: Work-Energy Theorem is a favorite. Conservation of mechanical energy and power calculations, sometimes with variable force scenarios, are often asked.
• Centre of Mass & Collisions: Calculation of COM for simple systems/continuous bodies. Different types of collisions (elastic, inelastic, perfectly inelastic) mainly focus on 1D collisions, applying momentum and KE conservation.
• Rotational Motion: Moment of inertia calculations (using theorems), torque-angular acceleration relation. Conservation of angular momentum is a key concept. Rolling motion on inclined planes (acceleration, friction) is a standard problem type.
• Gravitation: Variation of 'g' (height, depth, latitude). Escape and orbital velocities. Kepler's laws, especially $T^2 \propto r^3$. Energy of satellites.
• Properties of Matter: Direct formula application for stress, strain, Hooke's Law, and various moduli. Fluid statics (Pascal's law, Archimedes' principle, pressure). Fluid dynamics (continuity equation, Bernoulli's principle, Torricelli's law). Viscosity (Stokes' law, terminal velocity). Surface tension (excess pressure in bubbles/drops, capillary rise).

Revision Checklist (Mechanics & Rotational Motion)

Make sure you can:

• Define and differentiate all basic quantities (displacement, velocity, acceleration, work, energy, power, force, momentum, torque, angular momentum, moment of inertia).
• Apply all kinematic equations for constant linear and angular acceleration.
• Solve projectile motion problems for range, height, time of flight.
• Solve relative motion problems (rain-man, river-boat).
• Draw accurate Free Body Diagrams (FBDs) for NLM problems.
• Apply Newton's Laws correctly to connected bodies and systems with friction.
• Calculate impulse and apply the impulse-momentum theorem.
• Understand and apply the conservation of linear momentum.
• Calculate work done by constant and variable forces.
• Apply the Work-Energy Theorem.
• Differentiate between conservative and non-conservative forces.
• Apply the conservation of mechanical energy.
• Calculate power (average and instantaneous).
• Locate the center of mass for discrete systems and simple continuous bodies.
• Distinguish between elastic, inelastic, and perfectly inelastic collisions and apply conservation laws accordingly.
• Calculate moment of inertia for various shapes and use parallel/perpendicular axis theorems.
• Calculate torque and angular momentum.
• Apply Newton's second law for rotation ($\tau = I\alpha$).
• Apply the conservation of angular momentum.
• Calculate rotational and total kinetic energy (including rolling).
• Analyze rolling motion without slipping on inclined planes.
• Apply Newton's Law of Gravitation.
• Calculate variation of $g$ due to height, depth, and rotation.
• Calculate gravitational potential energy and potential.
• Calculate escape velocity and orbital velocity.
• Apply Kepler's Laws.
• Define and use various elastic moduli (Young's, Bulk, Shear) and Poisson's ratio.
• Apply concepts of fluid statics (Pascal's Law, Archimedes' Principle, pressure variation).
• Apply the equation of continuity and Bernoulli's Principle for fluid dynamics.
• Solve problems related to viscosity (Stokes' Law, terminal velocity).
• Solve problems related to surface tension (surface energy, excess pressure, capillarity).

Last-Day Strategy

1. Quick Scan: On the last day, quickly scan this formula sheet. Do NOT try to solve new problems.
2. Focus on Formulas: Read through every formula and concept. Mentally link them to their applications.
3. Units and Constants: Pay close attention to units of each quantity and values of important constants like G, g (Earth).
4. Key Diagrams: Briefly recall FBDs for common scenarios (pulleys, inclined planes), and velocity vectors for projectile/relative motion.
5. Distinctions: Revisit distinctions like conservative vs. non-conservative forces, elastic vs. inelastic collisions, static vs. kinetic friction.
6. Mnemonics: Use your memory tricks or the ones provided to solidify recall.
7. Confidence: Trust in your preparation. A calm mind performs best! Don't panic if you forget something, move on and come back later.
8. Sleep: Get adequate sleep. Being well-rested is more beneficial than last-minute cramming.

Good luck with your NEET preparation!