Mechanics

Learning Objectives

• Master SI units, dimensional analysis, and error propagation for accurate measurements
• Solve kinematics problems in 1D and 2D using equations of motion, graphs, and vector analysis
• Apply Newton's laws with free-body diagrams for systems involving friction, pulleys, and circular motion
• Analyse work-energy relationships, power, and conservation principles in mechanical systems
• Calculate centre of mass, moment of inertia, torque, and angular momentum for rotating bodies
• Derive gravitational field, potential, orbital and escape velocities using Kepler's & Newton's laws
• Understand stress-strain behaviour of solids and fluid statics, dynamics, and viscosity fundamentals

1. Units & Measurements

SI Units & Dimensions

Fundamental & Derived Units

The International System of Units (SI) defines seven fundamental quantities from which all other physical quantities are derived. Every measurement in physics ultimately traces back to these base units. The seven base quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. All other quantities (velocity, force, energy, etc.) are derived from these base units.

Supplementary units: radian (rad) for plane angle and steradian (sr) for solid angle. While these are dimensionless, they are essential for rotational and wave mechanics. Prefixes like milli (10⁻³), centi (10⁻²), kilo (10³), mega (10⁶), giga (10⁹) are used for multiples and sub-multiples.

Dimensional Analysis

Dimensions are expressed in terms of M (mass), L (length), T (time), A (current), K (temperature), mol (amount), cd (luminous intensity). The dimensional formula of a physical quantity is the expression of the base dimensions it depends upon. The principle of homogeneity states that only quantities with the same dimensions can be added, subtracted, or equated. This principle is the foundation of dimensional analysis and is used to check the correctness of physical equations and to derive relationships between quantities.

Significant Figures

Rules for counting significant figures: (1) All non-zero digits are significant. (2) Zeros between non-zero digits are significant. (3) Leading zeros are not significant. (4) Trailing zeros after decimal are significant. (5) In scientific notation N × 10^m, all digits in N are significant. For addition/subtraction, round to the least precise decimal place. For multiplication/division, round to the least number of significant figures. Examples: 5200 (2 significant figures, unless written as 5200. which has 4), 0.0045 (2 significant figures), 3.1416 (5 significant figures). Use scientific notation to avoid ambiguity: 5.2 × 10³ clearly has 2 significant figures.

Rounding Off

Rules for rounding off: (1) If the digit to be dropped is less than 5, the preceding digit is unchanged. (2) If the digit to be dropped is greater than 5, the preceding digit is increased by 1. (3) If the digit to be dropped is exactly 5, the preceding digit is increased by 1 if it is odd, unchanged if it is even (rule of even digit). Example: 3.45 rounds to 3.4 (4 is even), 3.35 rounds to 3.4 (3 is odd).

Error Analysis

Absolute Error = |Measured Value − True Value|. Mean Absolute Error = (Σ|Δx_i|)/n. Relative Error = Mean Absolute Error / Mean Value. Percentage Error = Relative Error × 100%. For a quantity Z = A^pB^q / C^r, the maximum relative error is ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C).

2. Kinematics

Motion in 1D

Equations of Motion (Constant Acceleration)

For motion with constant acceleration a, the three primary equations are:

• v = u + at
• s = ut + ½at²
• v² = u² + 2as

where u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement. These are valid only for uniformly accelerated motion along a straight line. They can also be expressed in vector form for 2D motion: v = u + at, r = ut + ½at².

Graphs in Kinematics

Graphical analysis is a powerful tool for solving kinematics problems. Three key graph types are used:

• Displacement–Time (s–t) graph: Slope = velocity. A straight line indicates constant velocity; a curved line indicates acceleration. The area under the curve has no physical meaning.
• Velocity–Time (v–t) graph: Slope = acceleration. Area under the graph = displacement (distance for speed–time). A horizontal line represents uniform velocity; a sloping line represents uniform acceleration.
• Acceleration–Time (a–t) graph: Area under the graph = change in velocity. Area = v_f − v_i.

These graphs are interconvertible: differentiate s–t to get v–t; differentiate v–t to get a–t. Integrate a–t to get v–t; integrate v–t to get s–t. For v–t graphs: when the graph crosses the time axis, the body reverses direction. NEET problems frequently ask: "Which v–t graph corresponds to this s–t graph?" or vice versa.

Derived formulas for motion under gravity: For free fall (a = g), the equations become v = u + gt, h = ut + ½gt², v² = u² + 2gh. For upward projection, g is taken as negative (a = −g). Time of ascent = time of descent = u/g. Maximum height = u²/2g. For a body dropped from rest, distance travelled in n^th second: S_n = u + a(2n − 1)/2 = g(2n − 1)/2. For vertically projected body, the speed at any height h is v = √(u² − 2gh). The ratio of distances travelled in successive seconds is 1:3:5:7:... for a body starting from rest.

Graphical Analysis

Slope of s-t graph gives velocity. Slope of v-t graph gives acceleration. Area under v-t graph gives displacement (with sign). Area under a-t graph gives change in velocity. For non-uniform acceleration, the slope of the v-t graph at any point gives instantaneous acceleration. Key points for graph interpretation:

• In a v-t graph, a horizontal line means constant velocity (a = 0)
• A straight line with positive slope means constant positive acceleration
• A curved line means non-uniform acceleration
• The area between v-t graph and time axis = displacement; if the graph goes below the axis, that area represents negative displacement
• The magnitude of displacement = area under speed-time graph (always positive)

Motion in 2D

Vectors in Kinematics

Position vector r = xᴋ + yᴌ. Velocity v = dr/dt = v_xᴋ + v_yᴌ. Acceleration a = dv/dt = a_xᴋ + a_yᴌ. For uniform acceleration: r = r₀ + ut + ½at². Component-wise analysis is key to solving 2D problems. The motion in x and y directions are independent of each other.

Vector Addition and Resolution

Vectors follow the triangle and parallelogram laws of addition. For two vectors A and B with angle θ between them, the magnitude of their resultant R = √(A² + B² + 2AB cosθ), and the direction is given by tanφ = B sinθ/(A + B cosθ). Resolution of a vector into components: A_x = A cosθ, A_y = A sinθ (where θ is measured from the x-axis). Unit vectors along the axes: ᴋ, ᴌ, ᴍ. The dot product A·B = AB cosθ = A_xB_x + A_yB_y + A_zB_z. The cross product A × B = AB sinθ with direction given by the right-hand rule. Cross product magnitude equals the area of the parallelogram formed by the vectors. For NEET, focus on vector addition (co-planar vectors) and resolution into rectangular components using trigonometric functions.

Relative velocity in 2D: v_AB = v_A − v_B (vector subtraction). For two bodies moving at angles, relative velocity magnitude = √(v_A² + v_B² − 2v_Av_Bcosθ). The shortest distance between two moving bodies can be found using relative velocity concepts.

Projectile Motion

A projectile launched with initial velocity u at angle θ with horizontal follows a parabolic trajectory under constant gravitational acceleration g (ignoring air resistance). The horizontal component of velocity remains constant throughout the motion, while the vertical component changes due to gravity. The motion can be analysed by treating the horizontal and vertical components independently.

• Horizontal velocity: u_x = u cosθ (constant throughout)
• Vertical velocity: u_y = u sinθ − gt (changes due to g)
• Time of flight: T = 2u sinθ / g
• Maximum height: H = u² sin²θ / 2g
• Range: R = u² sin2θ / g
• Equation of trajectory: y = x tanθ − gx² / 2u² cos²θ

Projectile from a height: If a projectile is fired from a height h above ground level, the equation of trajectory method is used. Time of flight is obtained by solving for y = −h in the vertical motion equation. Range = u cosθ × T. For horizontal projection (θ = 0) from height h: T = √(2h/g), R = u × √(2h/g). For a projectile fired horizontally from a moving platform (e.g., an aeroplane at altitude h with speed u), the bomb's initial horizontal velocity equals the plane's speed. The bombs fall in a parabolic path and are always vertically below the plane (neglecting air resistance). The distance from the target = u × √(2h/g).

Projectile on an inclined plane: When a projectile is fired up an incline (angle β) with velocity u at angle θ with the incline: range up the incline R = 2u²cosθsin(θ+β)/gcos²β. Time of flight T = 2u sin(θ+β)/(g cosβ). Maximum range on an inclined plane occurs at θ = 45° − β/2. For a projectile fired down an incline, the angle for maximum range is θ = 45° + β/2. These formulas are useful for NEET, but it is often faster to resolve the acceleration due to gravity along and perpendicular to the incline.

Relative Motion

Velocity of A relative to B: v_AB = v_A − v_B. For relative velocity in 2D, vector subtraction applies. In rain-man problems, the relative velocity of rain w.r.t. man determines the angle at which the man should hold his umbrella. In river-boat problems, the velocity of the boat relative to ground is the vector sum of boat velocity relative to water and water velocity relative to ground.

3. Laws of Motion

Newton's Laws

First Law (Law of Inertia): A body continues in its state of rest or uniform motion unless acted upon by an external unbalanced force. Inertia is the property of mass that resists change in motion. Greater mass means greater inertia. Examples: when a bus suddenly stops, passengers lurch forward (due to inertia of motion); when a bus suddenly starts, passengers fall backward (inertia of rest).

Second Law: The rate of change of momentum is directly proportional to the applied force and takes place in the direction of the force. Mathematically, F = dp/dt = d(mv)/dt = ma (for constant mass). The SI unit of force is newton (N) = kg·m/s². The second law also gives the concept of impulse: Impulse J = FΔt = Δp = m(v − u). The impulse-momentum theorem is especially useful for problems involving varying forces and collisions where the time of contact is known.

Third Law: Every action has an equal and opposite reaction. F_AB = −F_BA. Action-reaction pairs act on different bodies, never on the same body. Examples: a gun recoils when a bullet is fired; a rocket propulsion works by expelling gases downward; a man walking on ground pushes the ground backward and the ground pushes him forward.

Analysis of Bouncing Ball

The motion of a ball bouncing off a floor is a rich problem combining free fall, impact, and coefficient of restitution. Key stages: (1) Ball dropped from height h: speeds up to v = √(2gh) just before impact. (2) During impact (Δt ≈ 0.01 s): large impulsive force acts upward; impulse J = m(v' + v) = m(v + ev). (3) Ball rebounds with speed v' = ev = e√(2gh) and rises to height h' = e²h. (4) The total time for n bounces: total time = √(2h/g)[1 + 2e + 2e² + ...] = √(2h/g)(1 + e)/(1 − e). (5) The total distance travelled before stopping = h + 2e²h + 2e⁴h + ... = h(1 + e²)/(1 − e²). These geometric series results are frequently tested in NEET and other competitive exams.

Free Body Diagrams

A free body diagram (FBD) isolates a body and shows all forces acting on it. Steps: (1) Identify the system/body of interest. (2) Draw all external forces (gravity mg downwards, normal reaction N perpendicular to surface, tension T along the string away from the body, friction f opposing relative motion, applied forces). (3) Resolve forces into components along convenient axes (usually along the direction of acceleration and perpendicular to it). (4) Apply ΣF = ma along each axis. For equilibrium (a = 0), ΣF = 0 in all directions.

Connected Motion (Pulley Systems)

For systems of connected bodies, treat the entire system to find acceleration, then isolate individual bodies to find tension. For an Atwood machine (two masses m₁ and m₂ connected by a string over a frictionless, massless pulley): a = (m₁ − m₂)g / (m₁ + m₂) and T = 2m₁m₂g / (m₁ + m₂). For systems with multiple blocks and pulleys, write constraint equations relating the accelerations. The number of independent constraint equations equals the number of strings.

Mass on incline connected to hanging mass: If mass m₁ is on a smooth incline at angle θ and mass m₂ is hanging vertically, connected by a string over a pulley at the top: a = (m₂g − m₁g sinθ)/(m₁ + m₂). If the incline is rough with coefficient μ, subtract kinetic friction from the numerator: a = (m₂g − m₁g sinθ − μm₁g cosθ)/(m₁ + m₂). Tension T = m₂(g − a) = m₁(a + g sinθ + μg cosθ). For multiple pulley systems (block and tackle), the mechanical advantage equals the number of rope segments supporting the load.

Pseudo Forces (Non-Inertial Frames)

When analysing motion from a non-inertial (accelerated) frame of reference, a pseudo force F_p = −ma₀ must be applied to each body of mass m, where a₀ is the acceleration of the frame. This is useful for problems involving accelerating lifts, vehicles, or rotating reference frames. For an accelerating lift: apparent weight = m(g ± a). If lift accelerates upward, apparent weight = m(g + a) (heavier feeling). If lift accelerates downward, apparent weight = m(g − a) (lighter feeling). If the cable breaks (free fall), a = g, apparent weight = 0 (weightlessness).

Galilean Relativity: The laws of mechanics are the same in all inertial frames. The velocity of a body in frame S' moving with constant velocity v relative to frame S is v' = v − V. Acceleration is invariant (same in all inertial frames): a' = a. This is why a ball dropped from a moving train appears to fall vertically to an observer on the train, but follows a parabolic path to an observer on the ground.

Rotating frames and centrifugal force: In a rotating reference frame with angular velocity ω, pseudo forces experienced include: centrifugal force F_cf = mω²r (away from centre) and Coriolis force F_cor = −2m(ω × v) (deflects moving objects sideways). The Coriolis force explains the direction of cyclones (counterclockwise in Northern hemisphere, clockwise in Southern) and the deviation of long-range projectiles. Centrifugal force affects the effective value of g at different latitudes.

Lami's Theorem (Equilibrium of Concurrent Forces)

If three concurrent forces F₁, F₂, F₃ acting at a point keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces: F₁/sinα = F₂/sinβ = F₃/sinγ, where α is the angle between F₂ and F₃, β between F₁ and F₃, and γ between F₁ and F₂. This theorem is very useful for solving problems involving strings, springs, and walls in equilibrium without resolving into components.

Friction

Friction is a contact force that opposes relative motion or tendency of relative motion between surfaces in contact.

Typically: μ_s > μ_k > μ_r. N is the normal reaction force. Friction is independent of the area of contact (to first approximation). The angle of repose α = tan⁻¹(μ_s) is the maximum angle of an inclined plane before a block slips. For an inclined plane at angle θ:

• If tanθ < μ_s: block remains at rest (friction = mg sinθ)
• If tanθ = μ_s: block is on verge of slipping (limiting friction)
• If tanθ > μ_s: block slides down with acceleration a = g(sinθ − μ_k cosθ)

For a block pushed up an incline with force F parallel to the incline: net acceleration depends on whether F exceeds the sum of mg sinθ and friction. For a block on a rough horizontal surface pulled at angle θ above horizontal: N = mg − F sinθ, so friction is reduced, making it easier to move the block.

Circular Motion

For uniform circular motion, the speed is constant but the velocity direction changes continuously, producing centripetal acceleration directed toward the centre. Centripetal acceleration a_c = v²/r = ω²r. Centripetal force F_c = mv²/r = mω²r. Angular velocity ω = 2π/T = 2πf, where T is time period and f is frequency. The centripetal force is not a separate force but is provided by tension, gravity, friction, or normal reaction depending on the situation.

Friction in Circular Motion

For a vehicle negotiating a flat circular curve, the centripetal force is provided entirely by friction. The maximum safe speed without skidding is v_max = √(μ_srg), where μ_s is the coefficient of static friction between tyres and road. For a banked curve (angle θ), the optimum speed requiring no friction is v_o = √(rg tanθ). If friction is also present, the maximum safe speed on a banked curve increases: v_max = √(rg(μ_s + tanθ)/(1 − μ_s tanθ)). In practice, roads are banked for the average expected speed, and friction handles deviations. In a "death well" (vertical circular wall), the minimum speed to avoid sliding down is v = √(rg/μ), where the normal reaction provides centripetal force and friction balances weight.

Motion in a Vertical Circle

A body moving in a vertical circle experiences both centripetal force and varying gravitational force. The tension or normal reaction varies at different points. For a mass m tied to a string whirled in a vertical circle of radius R:

• At topmost point (A): Minimum speed v_A = √(gR) for string tautness. T_A = mv_A²/R − mg = 0 (at minimum speed).
• At lowest point (B): v_B = √(5gR) for completing the circle. T_B = mv_B²/R + mg = 6mg.
• At horizontal position (C): v_C = √(3gR). T_C = mv_C²/R = 3mg.

Critical condition: v_B ≥ √(5gR) for complete vertical circle. If v_B < √(2gR), the mass oscillates about the lowest point. If √(2gR) < v_B < √(5gR), the mass leaves the circle between the horizontal and topmost positions and follows a parabolic path. For a bead on a rigid circular wire, the bead can complete the circle at any non-zero speed because the wire provides radial support (unlike a string which can only provide tension). The difference in tension between lowest and highest points is always 6mg, independent of speed, which is a useful NEET shortcut.

4. Work, Energy & Power

Work

Work done by a constant force: W = F·s = Fs cosθ, where θ is the angle between force and displacement vectors. Work is a scalar quantity and its SI unit is joule (J).

• Positive work: 0° ≤ θ < 90° (force aids motion; e.g., gravity when falling)
• Zero work: θ = 90° (force perpendicular to displacement; e.g., centripetal force, tension in a string for horizontal circular motion)
• Negative work: 90° < θ ≤ 180° (force opposes motion; e.g., friction, gravity when moving upward)

Work done by a variable force is given by the area under the force-displacement graph: W = ∫F·ds.

Work-Energy Theorem

Work done by net force equals change in kinetic energy: W_net = ΔKE = KE_f − KE_i = ½m(v_f² − v_i²). This is a powerful tool that avoids detailed kinematic analysis and can solve problems involving non-uniform acceleration.

Conservative vs Non-Conservative Forces

Conservative forces (gravity, spring force, electrostatic force): Work done is path-independent; work done around a closed path is zero; total mechanical energy is conserved. Δ(KE + PE) = 0. The work done by a conservative force equals the negative of the change in potential energy: W_c = −ΔPE.

Non-conservative forces (friction, air drag, viscous force): Work done depends on path; mechanical energy is not conserved. W_NC = ΔKE + ΔPE = change in total mechanical energy.

Power

Power is the rate of doing work: P = dW/dt = F · v (for constant force). SI unit: watt (W) = J/s. Average power = Total work / Total time. Instantaneous power = Fv cosθ. Power is also equal to the rate at which energy is transferred or converted.

Work-Energy Theorem in Rotational Motion

The work-energy theorem extends to rotational motion: work done by torque = change in rotational kinetic energy. W = ∫τdθ = Δ(½Iω²). For a body undergoing both translation and rotation, total work done by forces = change in total KE: W = Δ(½mv_cm² + ½I_cmω²). A classic problem: a yo-yo falling while unwinding string uses this principle to relate linear and angular motion. The power delivered by a torque P = τω.

Mechanical Advantage, Velocity Ratio & Efficiency

Simple machines (levers, pulleys, inclined planes, screws, wedges, gears) help lift heavy loads with less effort. Key parameters: Mechanical Advantage (MA) = Load/Effort. Velocity Ratio (VR) = Distance moved by effort / Distance moved by load. Efficiency η = MA/VR = Work output / Work input = Useful work done / Total work done. For an ideal (frictionless) machine, MA = VR and η = 1 (100%). In real machines, friction reduces efficiency. For a lever: MA = Load arm / Effort arm (for first-class lever like see-saw or crowbar). For a single fixed pulley: MA = 1 (only changes direction). For a single movable pulley: MA = 2. For a system of n pulleys (block and tackle): MA = n (ideal). For an inclined plane: MA = Length/Height = 1/sinθ. The screw jack has a very high MA = 2πL/p, where L = lever arm length and p = pitch of the screw. For NEET, focus on lever and pulley problems with efficiency.

Variable Mass Systems: Rocket Propulsion

For systems where mass changes with time (like a rocket ejecting fuel), Newton's second law is applied in its original form F = dp/dt = d(mv)/dt. The rocket equation (Tsiolkovsky equation) gives the velocity of a rocket at any instant: v = v₀ + u ln(m₀/m), where u is the exhaust velocity of gases relative to the rocket, m₀ is the initial total mass, and m is the mass at that instant. The thrust force on the rocket = u(dm/dt). For a rocket launched vertically, net acceleration a = (u dm/dt)/m − g. The total impulse delivered = u(m₀ − m). Multistage rockets use staging to drop empty fuel tanks, improving the mass ratio and achieving higher final velocities.

5. Rotational Motion

Kinematics of Rotational Motion

Rotational motion is described by angular displacement (θ), angular velocity (ω = dθ/dt), and angular acceleration (α = dω/dt = d²θ/dt²). For constant angular acceleration, the equations are completely analogous to linear kinematics: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ. The relationship between linear and angular quantities for a point at distance r from the axis: s = rθ (arc length), v = rω (tangential speed), a_t = rα (tangential acceleration), a_c = v²/r = rω² (centripetal/radial acceleration). Total linear acceleration magnitude: a = √(a_t² + a_c²). The direction of the total acceleration is not along the radius unless α = 0 (uniform circular motion). The angular velocity vector direction is given by the right-hand thumb rule.

Centre of Mass

For a system of particles, the centre of mass (COM) is the weighted average of positions: R_CM = (Σm_ir_i) / (Σm_i). For continuous bodies, R_CM = (∫r dm) / M. Velocity of COM: V_CM = (Σm_iv_i) / M. Acceleration of COM: A_CM = F_ext / M. In the absence of external forces, COM remains at rest or moves with constant velocity. The COM of a system of particles can lie outside the material of the body (e.g., a ring, a hollow sphere, or a boomerang). COM of common bodies: uniform rod = mid-point; triangle = centroid at h/3 from base; semicircular ring = 2R/π from centre; semicircular disc = 4R/3π from centre; solid hemisphere = 3R/8 from centre; hollow hemisphere = R/2 from centre.

Moment of Inertia

Moment of inertia I = Σm_ir_i² (discrete system) or I = ∫r² dm (continuous body). It measures rotational inertia, analogous to mass in linear motion. SI unit: kg·m². Two important theorems:

• Perpendicular Axis Theorem (for planar bodies): I_z = I_x + I_y, where z is perpendicular to the plane.
• Parallel Axis Theorem: I = I_CM + Mh², where h is the distance between the parallel axes.

Torque & Angular Momentum

Torque: τ = r × F, magnitude τ = rF sinθ. Direction is given by the right-hand rule (perpendicular to plane of r and F). Torque is the rotational analogue of force. The net external torque about a point equals the rate of change of angular momentum: τ_ext = dL/dt. For a system of particles, τ_ext = Iα (where α = dω/dt is angular acceleration).

Angular momentum: L = r × p = m(r × v). For rotation about a fixed axis: L = Iω. The direction of L is along the axis of rotation (right-hand rule). Conservation of angular momentum: if τ_ext = 0, then L = constant, i.e., I₁ω₁ = I₂ω₂. This explains why a spinning ice skater rotates faster when pulling arms in (decreasing I increases ω to keep L constant), why a diver tucks in to spin faster, and why a cat always lands on its feet.

Couple: A pair of equal, parallel, and opposite forces separated by distance d produces a torque (couple) of magnitude τ = Fd, independent of the point about which it is taken. A couple produces pure rotation without translation.

Equilibrium of rigid bodies: For a body to be in complete equilibrium, both net force and net torque must be zero: ΣF = 0 and Στ = 0 (about any point). This is used to solve problems involving ladders, beams, and see-saws.

Rolling Motion

For pure rolling without slipping: v = ωR. Total KE = KE_trans + KE_rot = ½mv² + ½Iω² = ½mv²(1 + I/mR²). The friction in pure rolling is static and does no work (point of contact is instantaneously at rest). For rolling down an incline, acceleration a = g sinθ / (1 + I/mR²). The object with smaller I/mR² reaches bottom first.

Centre of Mass Motion & Collisions

Velocity of centre of mass: v_cm = (m₁v₁ + m₂v₂ + ...)/(m₁ + m₂ + ...) = total momentum / total mass. In the absence of external forces, the velocity of the centre of mass remains constant (total momentum conserved). The centre of mass behaves as if all mass is concentrated there and all external forces act there. For two-body collisions, the centre of mass frame simplifies analysis significantly.

Collisions in Centre of Mass Frame: In the centre of mass frame, the total momentum is zero. For elastic collisions, the speeds of approach and separation are equal in the centre of mass frame: relative speed of approach = relative speed of separation. The coefficient of restitution e = relative speed of separation / relative speed of approach. For perfectly elastic collision, e = 1; for perfectly inelastic, e = 0; for partially elastic, 0 < e < 1.

6. Gravitation

Kepler's Laws

First Law (Law of Orbits): Each planet moves in an ellipse with the Sun at one focus. The semi-major axis a and eccentricity e define the orbit's shape. The eccentricity e = √(1 − b²/a²), where b is the semi-minor axis. For Earth's orbit, e ≈ 0.0167 (nearly circular).

Second Law (Law of Areas): The line joining the planet to the Sun sweeps equal areas in equal time intervals. dA/dt = L/2m = constant, where L is the angular momentum. This implies that the planet moves faster when closer to the Sun (perihelion) and slower when farther (aphelion). The areal velocity is constant: dA/dt = ½r²ω = constant.

Third Law (Law of Periods): The square of the orbital period is proportional to the cube of the semi-major axis. T² ∝ a³. For circular orbits around a central mass M: T² = (4π²/GM)r³. This is derived by equating centripetal force to gravitational force: GMm/r² = mω²r = m(2π/T)²r.

Types of Satellites

Geostationary satellites: Orbit in the equatorial plane with T = 24 h, matching Earth's rotation. Orbital radius r ≈ 42000 km (altitude ≈ 36000 km). They appear stationary from Earth. Used for communication, broadcasting, weather monitoring.

Low Earth Orbit (LEO) satellites: Altitude 200-2000 km, T ≈ 90 min, v ≈ 7.9 km/s. Used for remote sensing, spy satellites, International Space Station.

Polar satellites: Orbit passing over the poles at low altitude (800 km). Used for mapping, environmental monitoring, reconnaissance.

Universal Gravitation

Newton's law of universal gravitation: F = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N·m²/kg². Gravitational force is attractive, central (acts along the line joining the centres), and follows the inverse square law. It is the weakest of the four fundamental forces but dominates at astronomical scales. Gravitational force is conservative.

Gravitational Field of Continuous Distributions

For calculating the gravitational field due to extended objects, we use integration or known results. Key results for NEET:

• Field inside a uniform spherical shell: E = 0 everywhere inside the shell. Potential V = constant = −GM/R (where R = radius of shell). This is a very important result: a spherical shell exerts no gravitational force on a particle inside it.
• Field outside a spherical shell or solid sphere: E = GM/r² (as if all mass is concentrated at the centre).
• Field inside a uniform solid sphere: E = GMr/R³ (directly proportional to distance from centre). At centre, E = 0. Potential V = −GM(3R² − r²)/2R³.
• Field due to a uniform circular ring: On the axis at distance x from centre, E = GMx/(R² + x²)^3/2. Maximum field at x = R/√2.

These results are frequently tested in NEET in comparison-type questions (e.g., "the gravitational field inside a hollow sphere is...") and in problems requiring integration or application of Gauss's law for gravity analogously to electrostatics.

Gravitational Field & Potential

Gravitational field intensity E = F/m = GM/r² (directed towards the mass causing the field). It is a vector quantity.

Gravitational potential V = W/m = −GM/r (scalar). It is the work done per unit mass in bringing a test mass from infinity to that point.

Gravitational potential energy U = −GMm/r. Relationship between field and potential: E = −dV/dr. The negative sign indicates that the field points in the direction of decreasing potential. The superposition principle applies: total field/potential is the vector/scalar sum of individual contributions.

Escape & Orbital Velocity

Escape Velocity: Minimum velocity required for an object to escape Earth's gravitational field (reach infinity with zero residual speed): v_e = √(2GM/R) = √(2gR). For Earth: v_e ≈ 11.2 km/s. It is independent of the mass of the escaping object and depends only on the mass and radius of the planet.

Orbital Velocity: Velocity required for a satellite to orbit at radius r: v_o = √(GM/r). For a circular orbit just above Earth's surface, v_o ≈ 7.9 km/s. Relation: v_e = √2 × v_o (for same r). v_o decreases with increasing orbital radius.

Energy of a satellite: Total energy E = KE + PE = GMm/2r + (−GMm/r) = −GMm/2r (negative, indicating a bound state). To increase orbital radius, energy must be added to the satellite.

7. Elasticity

Stress & Strain

Stress = Deforming Force / Area (SI: N/m² or Pa). Stress is of three types: longitudinal (tensile/compressive), volumetric (bulk), and shear (tangential). Strain = Change in dimension / Original dimension (dimensionless). Within elastic limit, stress ∝ strain (Hooke's Law). The proportionality constant is the modulus of elasticity.

Elastic Moduli

Young's Modulus Y = Longitudinal Stress / Longitudinal Strain = (F/A) / (ΔL/L) = FL / AΔL. It measures resistance to length change. For a wire: ΔL = FL/AY. Spring constant of a wire: k = YA/L. For a composite wire made of two different materials joined end-to-end, the extension in each part is proportional to (L/Y) of that part. For parallel combination of wires, the net spring constant k = k₁ + k₂ and for series combination, 1/k = 1/k₁ + 1/k₂.

Bulk Modulus B = Volume Stress / Volume Strain = −ΔP / (ΔV/V). It measures resistance to volume change. Compressibility = 1/B. For gases, isothermal bulk modulus = P (for ideal gas at constant temperature), adiabatic bulk modulus = γP (where γ = C_p/C_v). For liquids, B is very large (liquids are nearly incompressible).

Shear Modulus (Rigidity Modulus) G = Shear Stress / Shear Strain = (F/A) / θ. It measures resistance to shape change. Only solids can sustain shear stress; fluids cannot. For a cubical block subjected to a tangential force on its top face, shear strain = x/h where x is the displacement of the top face and h is the height.

Stress-Strain Curve & Elastic Hysteresis

The stress-strain curve for a ductile material (e.g., mild steel) shows distinct regions: (1) Proportionality limit (Hooke's law is valid up to this point; stress ∝ strain), (2) Elastic limit (beyond which permanent deformation occurs; the material will not return to original shape when load is removed), (3) Yield point (a large strain increase for a very small increase in stress; the material appears to flow), (4) Ultimate tensile strength (the maximum stress the material can withstand; necking begins), (5) Breaking point (fracture occurs, also called the fracture point). For brittle materials (e.g., cast iron, glass), there is no significant plastic region; fracture occurs near the elastic limit. For elastomers (e.g., rubber), there is a large elastic region with no well-defined yield point.

Elastic hysteresis: The area between the loading and unloading stress-strain curves represents the energy dissipated as heat within the material. This is called elastic hysteresis. Materials with high hysteresis (e.g., rubber) are used in shock absorbers and vibration dampers. The elastic after-effect is the delay in regaining original shape after unloading.

Poisson's Ratio

When a body is stretched longitudinally, it contracts laterally (and vice versa). Poisson's ratio σ (or μ) = lateral strain / longitudinal strain = −(ΔD/D) / (ΔL/L). The negative sign compensates for the fact that lateral strain is opposite in sign to longitudinal strain. For most materials, σ is between 0 and 0.5. Cork has σ ≈ 0 (no lateral contraction), rubber has σ ≈ 0.5 (nearly incompressible), and steel has σ ≈ 0.3. The theoretical limits are −1 < σ < 0.5. Poisson's ratio relates the three elastic moduli: Y = 3B(1 − 2σ) and Y = 2G(1 + σ). These relations are useful in NEET to find one modulus given the others.

Elastic Potential Energy

When a wire is stretched, work done is stored as elastic potential energy. The energy stored in a stretched wire = ½ × Stress × Strain × Volume = ½ F × ΔL (since volume = A·L). Elastic energy per unit volume = ½ × Stress × Strain = ½ Y (Strain)². For a spring: U = ½kx². This energy is recoverable when the material is unloaded (within elastic limit). The energy density concept is important in NEET problems asking for the energy stored in a stretched wire or rod.

Applications: Bending of Beams & Torsion

Bending of beams: When a beam is loaded, it bends. The top surface contracts (compression) and the bottom surface extends (tension). The neutral axis (at the centre of the beam) experiences no strain. The bending moment M = YI_G/R, where I_G is the geometric moment of inertia of the cross-section and R is the radius of curvature. For a rectangular beam, I_G = bd³/12. The depression of a cantilever (beam fixed at one end and loaded at the other) δ = WL³/3YI_G. This explains why I-beams are stronger (most material is far from the neutral axis, maximising I_G).

Torsion of a cylinder: When a cylinder is twisted, shear stress develops. The torque required to twist a solid cylinder of radius R through angle φ over length L is τ = (πGR⁴φ)/(2L). The quantity (πR⁴/2) is called the polar moment of inertia. Torsion pendulums and drive shafts use this principle.

Elasticity in Structural Design

Understanding elasticity is crucial for designing safe structures. Factor of safety = Ultimate stress / Working stress. For bridges and buildings, a high factor of safety (3–5) is used. The choice of material depends on the application: steel has high Young's modulus (stiffness) for beams and columns; rubber has low Young's modulus for shock absorbers and seals; concrete has high compressive strength but low tensile strength, so steel reinforcement (RCC) is used. The elastic limit must never be exceeded in normal use; permanent deformation would compromise structural integrity. The bending of beams, deflection of bridges, and twisting of drive shafts all rely on elasticity principles. The maximum safe load a wire can support without exceeding the elastic limit is given by F_max = Stress_{elastic limit} × Area.

9. Oscillations (Simple Harmonic Motion)

SHM Basics

Simple Harmonic Motion is a periodic oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and directed opposite to it. Mathematically, F = −kx, where k is the force constant. This leads to the differential equation d²x/dt² = −ω²x, whose solution is x = A sin(ωt + φ) or x = A cos(ωt + φ').

Parameters of SHM

• Amplitude (A): Maximum displacement from mean position
• Angular frequency (ω): ω = 2π/T = 2πf
• Time period (T): T = 2π/ω = 2π√(m/k) for spring-mass system
• Frequency (f): f = 1/T = ω/2π
• Phase (φ): Initial phase determines the starting position

Velocity & Acceleration in SHM

Velocity: v = ±ω√(A² − x²). Maximum velocity (at mean position x = 0): v_max = ωA. Acceleration: a = −ω²x. Maximum acceleration (at extreme positions x = ±A): a_max = ω²A.

Energy in SHM

Total mechanical energy in SHM is constant: E = ½kA² = ½mω²A². Kinetic energy varies as KE = ½mω²(A² − x²). Potential energy varies as PE = ½mω²x². At mean position (x = 0): KE = E, PE = 0. At extreme positions (x = ±A): KE = 0, PE = E.

Simple Pendulum

For a simple pendulum with small amplitude (sinθ ≈ θ): T = 2π√(L/g). The period is independent of the mass of the bob and the amplitude (for small amplitudes). For a compound pendulum: T = 2π√(I/mgd).

Spring-Mass Systems

For a horizontal spring: T = 2π√(m/k). For a vertical spring: same formula; the equilibrium extension due to gravity does not affect the period. For springs in series: 1/k_eq = 1/k₁ + 1/k₂. For springs in parallel: k_eq = k₁ + k₂.

Damped & Forced Oscillations

Damped oscillations: When energy is dissipated (due to friction, air resistance, etc.), the amplitude of oscillation decreases over time. The damping force is proportional to velocity: F_d = −bv. The equation becomes m(d²x/dt²) + b(dx/dt) + kx = 0. For light damping, x = A₀e^(−bt/2m) sin(ω't + φ), where ω' = √(ω₀² − (b/2m)²). The amplitude decays exponentially.

Forced oscillations: When an external periodic force F₀ sin(ωt) is applied to an oscillator, it oscillates at the driving frequency ω. At resonance (ω = ω₀), the amplitude becomes maximum. This is used in tuning circuits, bridges (Tacoma Narrows Bridge collapse), and musical instruments. The sharpness of resonance is measured by the quality factor Q = ω₀/Δω, where Δω is the half-width of the resonance curve. A high Q indicates low damping and sharp resonance.

Other Oscillatory Systems

Torsional Pendulum: A disc suspended by a wire executes torsional oscillations. The restoring torque is τ = −Cθ, where C is the torsional constant. Time period T = 2π√(I/C). This is independent of amplitude for small twists. Used in torsion pendulums for timing and in the Cavendish experiment to measure G.

Physical Pendulum (Compound Pendulum): Any rigid body pivoted at a point not through its centre of mass oscillates as a pendulum. T = 2π√(I/mgd), where I = moment of inertia about the pivot, d = distance from pivot to centre of mass. The equivalent length of a simple pendulum L_eq = I/md. For a uniform rod of length L pivoted at one end, I = mL²/3, d = L/2, so T = 2π√(2L/3g).

LC Oscillations: In an LC circuit, charge oscillates with ω = 1/√(LC). This is a mechanical analogy of SHM where charge q ←→ displacement x, current I ←→ velocity v, inductance L ←→ mass m, 1/C ←→ spring constant k. NEET sometimes asks SHM analogy questions.

8. Fluid Mechanics

Pressure & Density

Pressure P = F/A. SI unit: pascal (Pa) = N/m². 1 atm = 1.013 × 10⁵ Pa = 760 mm of Hg = 101.3 kPa. In a static fluid, pressure at depth h: P = P₀ + ρgh, where P₀ is atmospheric pressure at the surface. Pressure is same at all points at the same depth in a static fluid, irrespective of shape of the container (hydrostatic paradox). Density ρ = m/V (kg/m³). Relative density (specific gravity) = ρ_substance / ρ_water. A barometer measures atmospheric pressure (using a mercury column), while a manometer measures gauge pressure (pressure relative to atmosphere). Absolute pressure = Gauge pressure + Atmospheric pressure.

Pascal's Law

Pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. Hydraulic lift: F₂/F₁ = A₂/A₁. This provides mechanical advantage: a small force on a small-area piston can lift a heavy load on a large-area piston. Hydraulic brakes and hydraulic jacks work on this principle.

Archimedes' Principle

The buoyant force on a body immersed in a fluid equals the weight of the displaced fluid: F_b = ρ_fluid V_immersed g. A body floats if its average density is less than the fluid density. Apparent weight = Actual weight − Buoyant force. Fraction of volume immersed = ρ_body / ρ_fluid. For a floating body, weight = buoyant force. A body sinks if its density exceeds that of the fluid.

Bernoulli's Theorem & Applications

For an ideal incompressible fluid in streamline (non-turbulent) flow: P + ½ρv² + ρgh = constant along a streamline. This is a statement of energy conservation for fluid flow. The constant is the total pressure. Important applications:

• Venturimeter: Measures flow speed using pressure difference at constriction. The flow rate Q = A₁A₂√(2gh)/(A₁² − A₂²).
• Atomiser / Sprayer: Perfume spray works by creating low pressure at the constriction, drawing liquid up from the reservoir.
• Aerofoil lift: Airplane wing shape causes higher velocity above the wing (longer path) and lower pressure, creating lift. F = (P_below − P_above)A.
• Magnus effect: Spinning ball in cricket/football curves due to pressure difference caused by unequal airspeeds on opposite sides.
• Torricelli's theorem: Efflux velocity from an orifice at depth h: v = √(2gh). This is a special case of Bernoulli with P₁ = P₂ = P_atm and v₁ ≈ 0 at the free surface.
• Blood flow and aneurysm: In a constricted artery, velocity is higher and pressure is lower (Venturi effect). An aneurysm (ballooning of artery) occurs where pressure is higher after the constriction.
• Pitot tube: Measures fluid flow velocity by comparing stagnation pressure and static pressure: v = √(2(P₀ − P)/ρ).

Torricelli's theorem in detail: The speed of efflux of a liquid from an orifice in a tank at depth h below the free surface is v = √(2gh). The horizontal range of the efflux stream hitting the ground below the tank bottom: if the orifice is at height y above the ground and the liquid level is H above the ground (so h = H − y), then time to fall t = √(2y/g) and range R = vt = √(2(H−y) × 2y) = 2√(y(H−y)). Maximum range occurs when y = H/2 (i.e., the orifice is at half the total height).

Viscosity

Viscous force between adjacent layers of fluid: F = ηA(dv/dx), where η is the coefficient of viscosity (SI: Pa·s). Stokes' law: viscous drag force on a sphere moving through a viscous fluid = 6πηrv. Terminal velocity of a sphere falling under gravity in a viscous fluid: v_t = 2r²(ρ − σ)g / 9η, where ρ = density of sphere, σ = density of fluid.

Streamline & Turbulent Flow

In streamline (laminar) flow, fluid particles follow smooth paths that do not cross each other, with the velocity at any point being constant in time. In turbulent flow, the fluid undergoes irregular fluctuations and mixing with eddies. The critical velocity v_c is the speed at which the transition from laminar to turbulent occurs. Reynolds number R_e = ρvD/η, where D is the diameter. Flow is laminar if R_e < 1000, turbulent if R_e > 2000, and transitional in between. Higher viscosity promotes laminar flow; higher velocity and larger diameter promote turbulent flow. The equation of continuity and Bernoulli's equation apply only to streamline flow of ideal (non-viscous, incompressible) fluids.

Surface Tension

Surface tension S = F/L (N/m). It arises due to cohesive forces between liquid molecules at the surface, creating a tendency to minimise surface area. The surface of a liquid behaves like a stretched elastic membrane under tension. Excess pressure inside a liquid drop = 2S/R. Inside a soap bubble (two surfaces) = 4S/R. Inside a liquid jet = S/R.

Capillary rise/fall: h = 2S cosθ / ρgr, where θ is the contact angle, r is the radius of the capillary tube. For water (θ ≈ 0°, cosθ ≈ 1), h = 2S/ρgr (capillary rise). This phenomenon causes water to rise in a narrow tube, in soil, and in filter paper. For mercury (θ ≈ 140°, cosθ is negative), capillary depression occurs. Factors affecting capillary rise: h ∝ T, h ∝ 1/ρ, h ∝ 1/r. In a capillary tube of insufficient length (L < h), the liquid will not overflow; instead the meniscus radius adjusts so that the liquid rises to the full height of the tube and the meniscus becomes hemispherical (contact angle changes).

Surface energy = S × ΔA, where ΔA is the change in surface area. This is the work required to increase the surface area by ΔA at constant temperature. For a liquid film with two surfaces (like a soap bubble), total work = 2S × ΔA. The surface tension decreases with increasing temperature; it becomes zero at the critical temperature. The surface tension of water at 20°C = 0.0727 N/m. Addition of soap reduces the surface tension of water, which helps in cleaning by improving wetting. The angle of contact determines whether a liquid wets the surface (θ < 90°) or does not wet the surface (θ > 90°).

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