Mechanics Cheat Sheet for NEET

This cheat sheet provides a comprehensive overview of key Mechanics concepts essential for the NEET examination. It covers fundamental principles, important formulas, and common pitfalls, designed to aid in quick revision and effective problem-solving strategies.

Key Concepts

1. Kinematics: Deals with the description of motion without considering its causes. Key terms include displacement, distance, speed, velocity, and acceleration. * Uniform Motion: Constant velocity, zero acceleration. * Non-Uniform Motion: Changing velocity, non-zero acceleration. * Projectile Motion: Motion under gravity, comprising horizontal uniform motion and vertical uniformly accelerated motion. * Relative Motion: Describing the motion of an object with respect to another moving object or reference frame.

2. Newton's Laws of Motion: * 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. * Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass ($F_{net} = ma$). * Third Law: For every action, there is an equal and opposite reaction.

3. Work, Energy, and Power: * Work: Product of force and displacement in the direction of the force. It's a scalar quantity. * Energy: The capacity to do work. Types include kinetic energy (due to motion) and potential energy (due to position or configuration). * Work-Energy Theorem: The net work done on an object equals the change in its kinetic energy. * Conservation of Mechanical Energy: In the absence of non-conservative forces (like friction), total mechanical energy (potential + kinetic) remains constant. * Power: The rate at which work is done or energy is transferred.

4. Rotational Motion: * Rotational Kinematics: Describes rotational motion using angular displacement, angular velocity, and angular acceleration, analogous to linear kinematics. * Torque: The rotational equivalent of force, causing angular acceleration. ($\vec{\tau} = \vec{r} \times \vec{F}$). * Moment of Inertia: The rotational equivalent of mass, representing resistance to angular acceleration. * Angular Momentum: The rotational equivalent of linear momentum. In a closed system, it is conserved in the absence of external torque. * Rolling Motion: A combination of translational and rotational motion.

5. Gravitation: * Newton's Law of Universal Gravitation: States that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers ($F = \frac{GMm}{r^2}$). * Gravitational Potential Energy: The work done to bring a mass from infinity to a point in a gravitational field. * Escape Velocity: The minimum speed an object needs to escape the gravitational field of a massive body. * Kepler's Laws: Describe planetary motion.

6. Simple Harmonic Motion (SHM): * A special type of periodic motion where the restoring force is directly proportional to the displacement and acts towards the equilibrium position. * Characterized by amplitude, time period, and frequency. Examples include mass-spring systems and simple pendulums.

7. Properties of Solids and Fluids: * Solids: Elasticity (Young's modulus, bulk modulus, shear modulus). * Fluids (Liquids & Gases): Pressure, density, Archimedes' Principle, Pascal's Principle, Bernoulli's Principle, viscosity, surface tension.

Formulas & Results

Kinematics

• Equations of Motion (constant acceleration):
  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 $n^{th}$ second)
• Projectile Motion:
  • Time of flight ($T$) = $\frac{2u \sin\theta}{g}$
  • Maximum height ($H$) = $\frac{u^2 \sin^2\theta}{2g}$
  • Range ($R$) = $\frac{u^2 \sin(2\theta)}{g}$ (Maximum range at $\theta = 45^\circ$)

Newton's Laws and Friction

• Newton's Second Law: $\vec{F}_{net} = m\vec{a}$
• Impulse: $\vec{J} = \vec{F}_{avg}\Delta t = \Delta\vec{p}$
• Conservation of Linear Momentum: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ (for collisions)
• Coefficient of Restitution ($e$): $e = \frac{\text{relative speed after collision}}{\text{relative speed before collision}}$ ($0 \leq e \leq 1$)
• Static Friction: $f_s \leq \mu_s N$
• Kinetic Friction: $f_k = \mu_k N$ (where $\mu_k < \mu_s$)
• Centripetal Force: $F_c = \frac{mv^2}{r} = m\omega^2r$

Work, Energy, Power

• Work Done: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$
• Kinetic Energy: $K = \frac{1}{2}mv^2$
• Gravitational Potential Energy: $U = mgh$ (near Earth's surface)
• Spring Potential Energy: $U_s = \frac{1}{2}kx^2$
• Power: $P = \frac{W}{t} = \vec{F} \cdot \vec{v}$
• Work-Energy Theorem: $W_{net} = \Delta K$
• Conservation of Mechanical Energy: $K_i + U_i = K_f + U_f$ (if only conservative forces act)

Rotational Motion

• Angular Kinematics (constant angular acceleration):
  1. $\omega = \omega_0 + \alpha t$
  2. $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
  3. $\omega^2 = \omega_0^2 + 2\alpha\theta$
• Relation between Linear and Angular Quantities:
  • $v = r\omega$
  • $a_t = r\alpha$ (tangential acceleration)
  • $a_c = r\omega^2 = \frac{v^2}{r}$ (centripetal acceleration)
• Torque: $\vec{\tau} = \vec{r} \times \vec{F}$; $\tau = I\alpha$
• Moment of Inertia ($I$): $I = \sum m_i r_i^2$. For common shapes (e.g., ring $MR^2$, disc $\frac{1}{2}MR^2$, sphere $\frac{2}{5}MR^2$)
• Parallel Axis Theorem: $I = I_{CM} + Md^2$
• Perpendicular Axis Theorem: $I_z = I_x + I_y$ (for planar bodies)
• Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$
• Angular Momentum ($\vec{L}$): $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$
• Conservation of Angular Momentum: $I_1\omega_1 = I_2\omega_2$

Gravitation

• Gravitational Force: $F = \frac{GMm}{r^2}$
• Acceleration due to Gravity: $g = \frac{GM}{R^2}$ (on Earth's surface)
• Variation of $g$ with height: $g' = g \left(1 - \frac{2h}{R}\right)$ (for $h \ll R$)
• Variation of $g$ with depth: $g' = g \left(1 - \frac{d}{R}\right)$
• Gravitational Potential Energy: $U = -\frac{GMm}{r}$
• Gravitational Potential: $V = -\frac{GM}{r}$
• Escape Velocity: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$
• Orbital Velocity: $v_o = \sqrt{\frac{GM}{r}}$ (for orbit radius $r$)
• Kepler's Third Law: $T^2 \propto r^3$ or $\frac{T^2}{r^3} = \frac{4\pi^2}{GM}$

SHM

• Displacement: $x(t) = A \sin(\omega t + \phi)$ or $A \cos(\omega t + \phi)$
• Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (for spring), $\omega = \sqrt{\frac{g}{L}}$ (for simple pendulum)
• Time Period: $T = \frac{2\pi}{\omega}$
• Velocity: $v(t) = A\omega \cos(\omega t + \phi) = \pm\omega\sqrt{A^2 - x^2}$
• Acceleration: $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$
• Total Energy: $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2A^2$ (constant)

Properties of Matter

• Young's Modulus ($Y$): $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$
• Bulk Modulus ($B$): $B = \frac{\text{Pressure}}{\text{Volumetric Strain}} = \frac{-\Delta P}{\Delta V/V}$
• Viscosity Force (Stokes' Law): $F = 6\pi\eta rv$
• Surface Tension ($S$ with $\gamma$ also used): $F = S \times L$ (force per unit length)
• Pressure: $P = \frac{F}{A}$; $P = h\rho g$
• Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
• Equation of Continuity: $A_1v_1 = A_2v_2 = \text{constant}$

Worked Examples

1. Kinematics A particle starts from rest and moves with an acceleration of $2 \text{ m/s}^2$ for $10 \text{ s}$. It then moves with constant velocity for $5 \text{ s}$ and finally decelerates at $4 \text{ m/s}^2$ until it comes to rest. Calculate the total distance covered.

• Phase 1 (Acceleration):
  • $u = 0$, $a = 2 \text{ m/s}^2$, $t = 10 \text{ s}$
  • $s_1 = ut + \frac{1}{2}at^2 = 0 \cdot 10 + \frac{1}{2} \cdot 2 \cdot (10)^2 = 100 \text{ m}$
  • Final velocity $v_1 = u + at = 0 + 2 \cdot 10 = 20 \text{ m/s}$
• Phase 2 (Constant Velocity):
  • $v = 20 \text{ m/s}$, $t = 5 \text{ s}$
  • $s_2 = v \cdot t = 20 \cdot 5 = 100 \text{ m}$
• Phase 3 (Deceleration):
  • $u = 20 \text{ m/s}$, $v = 0$, $a = -4 \text{ m/s}^2$
  • $v^2 = u^2 + 2as_3 \implies 0^2 = (20)^2 + 2(-4)s_3$
  • $0 = 400 - 8s_3 \implies 8s_3 = 400 \implies s_3 = 50 \text{ m}$
• Total Distance: $S_{total} = s_1 + s_2 + s_3 = 100 + 100 + 50 = 250 \text{ m}$.

2. Work, Energy & Power A particle of mass $m=2 \text{ kg}$ is moved from a point $A(1, 2, 3) \text{ m}$ to $B(3, 4, 1) \text{ m}$ by a force $\vec{F} = (2\hat{i} + 3\hat{j} - \hat{k}) \text{ N}$. Calculate the work done by the force.

• Displacement Vector: $\vec{d} = \vec{r}_B - \vec{r}_A = (3-1)\hat{i} + (4-2)\hat{j} + (1-3)\hat{k}$
  • $\vec{d} = (2\hat{i} + 2\hat{j} - 2\hat{k}) \text{ m}$
• Work Done: $W = \vec{F} \cdot \vec{d}$
  • $W = (2\hat{i} + 3\hat{j} - \hat{k}) \cdot (2\hat{i} + 2\hat{j} - 2\hat{k})$
  • $W = (2)(2) + (3)(2) + (-1)(-2)$
  • $W = 4 + 6 + 2 = 12 \text{ J}$

3. Gravitation Calculate the escape velocity for a planet whose radius is $6400 \text{ km}$ and acceleration due to gravity is $9.8 \text{ m/s}^2$.

• Given: $R = 6400 \text{ km} = 6.4 \times 10^6 \text{ m}$, $g = 9.8 \text{ m/s}^2$
• Escape Velocity Formula: $v_e = \sqrt{2gR}$
• Substitution: $v_e = \sqrt{2 \times 9.8 \text{ m/s}^2 \times 6.4 \times 10^6 \text{ m}}$
  • $v_e = \sqrt{19.6 \times 6.4 \times 10^6} = \sqrt{125.44 \times 10^6}$
  • $v_e \approx 11.2 \times 10^3 \text{ m/s} = 11.2 \text{ km/s}$

Common Mistakes

• Confusing speed and velocity, or distance and displacement: Remember velocity and displacement are vector quantities, having both magnitude and direction.
• Incorrectly applying sign conventions: Especially for acceleration, displacement, and forces in one-dimensional motion or projectile motion.
• Not converting units to SI before calculations: This is a very frequent source of error. Always work in meters, kilograms, seconds.
• Neglecting friction or air resistance: Unless stated, assume ideal conditions, but be aware of when these forces should be considered.
• Misunderstanding Work-Energy Theorem for non-conservative forces: The net work done is the change in kinetic energy whether conservative or non-conservative forces are present. Only conservative forces allow for the strict conservation of mechanical energy.
• Incorrectly using parallel/perpendicular axis theorems: Ensure the conditions for their application are met.
• Applying conservation laws (momentum, energy, angular momentum) in non-isolated systems: Conservation laws only apply when there are no external forces or torques (or they cancel out).
• Forgetting vector nature of torque and angular momentum: Use the right-hand rule for direction.
• Confusing angular velocity ($\omega$) with frequency ($\nu$ or $f$): $\omega = 2\pi f$.

Quick Revision Checklist

• Can I confidently use kinematic equations for constant acceleration?
• Do I know the components of velocity and acceleration at different points in projectile motion?
• Can I draw free-body diagrams accurately and apply Newton's second law?
• Am I clear on the difference between static and kinetic friction?
• Can I calculate work done by a constant or variable force?
• Do I know the formulas for different forms of energy and the work-energy theorem?
• Can I apply the principle of conservation of mechanical energy?
• Do I understand the concepts of impulse and momentum, and impulse-momentum theorem?
• Can I solve problems involving collisions using conservation of momentum and coefficient of restitution?
• Am I familiar with centripetal force and its applications (e.g., banking of roads)?
• Can I relate linear and angular kinematic quantities ($v=r\omega$, $a_t=r\alpha$)?
• Can I calculate torque and angular momentum?
• Do I know the moment of inertia for standard objects and how to use parallel/perpendicular axis theorems?
• Can I apply the conservation of angular momentum?
• Do I know Newton's Law of Gravitation and the variation of 'g' with height/depth?
• Can I calculate escape velocity and orbital velocity?
• Am I comfortable with the equations for SHM (displacement, velocity, acceleration, energy)?
• Do I know the formulas for Young's modulus, surface tension, and Bernoulli's principle?