Topic overview
- Detailed exploration of Newton's laws of motion, conservation principles, universal gravitation, and rotational dynamics.
- Rigorous examination-oriented study notes featuring analytical concepts, formulas, derivations, and common conceptual traps.
- Includes exhaustive concept maps, step-wise worked examples, targeted MCQs, assertion-reason items, and rapid-revision summaries.
Concept 1 · foundation
Kinematics and Equations of Motion
Kinematics describes the motion of objects without considering the forces causing the motion, utilizing position, velocity, and acceleration.
Prerequisites
- Basic understanding of scalar and vector quantities.
- Fundamental units and dimensions in the SI system.
- Graphical representation of straight lines and slopes.
Definitions and scope
- Distance is the total path length covered by a moving object, which is a scalar quantity.
- Displacement is the shortest straight-line vector distance from the initial position to the final position.
- Speed is the rate of change of distance, while velocity is the rate of change of displacement.
- Acceleration is defined as the rate of change of velocity with respect to time.
Step-by-step explanation
- Uniform motion implies constant velocity and zero acceleration, where distance increases linearly with time.
- Non-uniform motion involves changing velocity, requiring the concept of instantaneous velocity via calculus: $v = \frac{ds}{dt}$.
- Instantaneous acceleration is given by the second derivative of displacement: $a = \frac{d^2s}{dt^2}$.
- The first kinematic equation relates final velocity, initial velocity, acceleration, and time: $v = u + at$.
- The second kinematic equation gives displacement under constant acceleration: $s = ut + \frac{1}{2}at^2$.
- The third kinematic equation connects velocities, acceleration, and displacement without time: $v^2 = u^2 + 2as$.
- Distance traveled in the $n$-th second is given by $S_n = u + \frac{a}{2}(2n - 1)$.
- Motion under gravity represents a special case of 1D motion where acceleration $a$ is replaced by acceleration due to gravity $g$ ($9.8 \text{ m/s}^2$ downwards).
Key points
- Displacement can be zero or negative, whereas distance is always positive.
- Slope of a displacement-time graph gives velocity.
- Slope of a velocity-time graph gives acceleration.
- Area under a velocity-time graph represents displacement.
- Area under an acceleration-time graph represents change in velocity.
- Free-fall implies an initial velocity of zero when dropped from rest.
Worked examples
- A car starts from rest and accelerates uniformly at $2 \text{ m/s}^2$ for $5 \text{ seconds}$. Calculate the final velocity and total distance covered. Solution: $u=0$, $a=2$, $t=5$. $v = u + at = 0 + (2)(5) = 10 \text{ m/s}$. $s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2}(2)(5^2) = 25 \text{ meters}$.
- A ball is thrown vertically upwards with an initial velocity of $20 \text{ m/s}$. Find the maximum height reached ($g = 10 \text{ m/s}^2$). Solution: At max height, $v = 0$. Using $v^2 = u^2 - 2gh$, $0 = (20)^2 - 2(10)h implies 20h = 400 implies h = 20 \text{ meters}$.
Applications and connections
- Analyzing vehicular traffic movement and braking distances.
- Calculating projectile trajectories in ballistics and sports.
- Designing safety mechanisms in elevators and amusement park rides.
Advanced points, exceptions and limitations
- In variable acceleration, kinematic equations cannot be used directly; integration of differential equations is mandatory.
- Relative motion in one and two dimensions requires vector subtraction of velocities: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$.
- Frame of reference dictates whether an observer perceives acceleration (inertial vs. non-inertial frames).
Additional examples
- A train moving on a straight track at constant speed.
- A stone dropped freely from a high tower.
- An athlete running on a 100m straight track.
Important distinctions
- Distance is scalar; displacement is a vector.
- Speed is magnitude of velocity; velocity includes direction.
- Average speed can never be zero for moving objects, whereas average displacement can be zero for a round trip.
Common misconceptions
- Thinking that zero velocity at an instant means acceleration is also zero (e.g., a ball at its highest point has $v=0$ but $g \neq 0$).
- Confusing distance with magnitude of displacement when the path is curved.
- Assuming acceleration is always in the direction of motion.
Exam focus
- Memorize the formula for distance traveled in the $n$-th second.
- Practice graph interpretation questions frequently asked in UP PGT.
- Remember sign conventions for upward and downward motion under gravity.
- Derivation and application of kinematic equations via calculus.
- Analysis of relative velocity in rain-man and river-swimmer problems.
Concept 2 · core
Newton's Laws of Motion and Momentum
Newton's laws form the cornerstone of classical mechanics, linking force, mass, inertia, and momentum.
Prerequisites
- Concept of vectors and vector addition.
- Understanding of mass versus weight.
- Basic calculus and algebra.
Definitions and scope
- Linear momentum ($\vec{p}$) is the product of mass and velocity: $\vec{p} = m\vec{v}$.
- Force is an external push or pull that changes or tends to change the state of rest or uniform motion of a body.
- Inertia is the inherent property of a body to resist any change in its state of rest or motion.
- Impulse is the large force acting for a very short duration, equal to the change in momentum.
Step-by-step explanation
- Newton's First Law (Law of Inertia) states that a body continues in its state of rest or uniform motion in a straight line unless compelled by an external force.
- Inertia is of three types: inertia of rest, inertia of motion, and inertia of direction.
- Newton's Second Law states that the rate of change of linear momentum of a body is directly proportional to the applied external force and takes place in the direction in which the force acts.
- Mathematical formulation of Second Law: $\vec{F} = \frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} = m\vec{a}$ (for constant mass).
- Newton's Third Law states that to every action, there is always an equal and opposite reaction.
- Action and reaction forces act on two different bodies, hence they do not cancel each other.
- Law of Conservation of Linear Momentum states that if no external force acts on a system of bodies, the total linear momentum of the system remains constant.
- Impulse-Momentum Theorem states that impulse equals change in momentum: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$.
Key points
- First law defines inertial frames of reference.
- Second law gives the quantitative measure of force ($F = ma$).
- Third law explains rocket propulsion and recoil of a gun.
- Momentum is a vector quantity measured in $\text{kg}\cdot\text{m/s}$.
- Unit of force in SI is Newton (N), and in CGS is dyne ($1 \text{ N} = 10^5 \text{ dynes}$).
- Apparent weight in a lift depends on upward or downward acceleration.
Worked examples
- A force of $10 \text{ N}$ acts on a body of mass $2 \text{ kg}$ for $3 \text{ seconds}$ starting from rest. Find the final momentum. Solution: $F = 10 \text{ N}$, $t = 3 \text{ s}$. Impulse $J = F \times t = 10 \times 3 = 30 \text{ N}\cdot\text{s}$. Since Impulse = Change in momentum, final momentum = $30 \text{ kg}\cdot\text{m/s}$.
- A bullet of mass $20 \text{ g}$ is fired horizontally with a speed of $150 \text{ m/s}$ from a pistol of mass $2 \text{ kg}$. Calculate the recoil velocity of the pistol. Solution: Initial momentum = 0. By conservation of momentum, $m_b v_b + m_p v_p = 0 implies (0.02)(150) + (2)v_p = 0 implies 3 + 2v_p = 0 implies v_p = -1.5 \text{ m/s}$.
Applications and connections
- Rocket propulsion in outer space based on momentum conservation.
- Cushioning effect in sports (e.g., catching a cricket ball by pulling hands back).
- Design of vehicular crumple zones to increase collision time and reduce impact force.
Advanced points, exceptions and limitations
- For variable mass systems (like burning rocket fuel), Newton's second law is written as $\vec{F} = m\frac{d\vec{v}}{dt} + v_{rel}\frac{dm}{dt}$.
- In non-inertial frames, pseudo forces (like centrifugal or inertial forces) must be introduced to apply Newton's laws.
- Constraints in connected bodies (pulleys, Atwood machine) require resolving tension and acceleration components.
Additional examples
- Dusting a carpet by beating it with a stick (inertia of rest).
- A passenger falling forward when a bus stops abruptly (inertia of motion).
- Recoil of a gun when fired.
Important distinctions
- Inertia is a qualitative property (mass is its measure); momentum is a quantitative dynamic property ($mv$).
- Mass is scalar; momentum is a vector.
- Action and reaction act on different bodies; balanced forces act on the same body.
Common misconceptions
- Believing action and reaction forces cancel each other out.
- Thinking that a force is required to keep an object moving at a constant velocity in outer space.
- Confusing mass with weight.
Exam focus
- Numerical problems on connected bodies over pulleys.
- Apparent weight calculations in elevators accelerating up or down.
- Derivations of rocket propulsion equation in PGT exams.
- Conservation of momentum in collisions (elastic and inelastic).
- Direction and magnitude of impulsive forces.
Concept 3 · core
Friction and Its Applications
Friction is the opposing contact force that arises when surfaces slide or tend to slide relative to each other.
Prerequisites
- Concept of normal reaction force.
- Resolution of forces into components on an inclined plane.
- Newton's laws of motion.
Definitions and scope
- Friction is the tangential force between two bodies in contact that opposes relative motion.
- Static friction ($f_s$) is the friction acting between surfaces when they are at rest relative to each other, up to a maximum limit.
- Limiting friction ($f_{s,max}$) is the maximum static friction before a body starts moving.
- Kinetic or sliding friction ($f_k$) is the friction acting when surfaces are in relative sliding motion.
Step-by-step explanation
- Origin of friction lies in atomic and molecular interactions and interlocking of microscopic surface irregularities.
- Laws of limiting friction state that limiting friction is directly proportional to the normal reaction ($N$): $f_{s,max} = \mu_s N$.
- Coefficient of static friction ($\mu_s$) is a dimensionless constant depending on the nature of contacting surfaces.
- Kinetic friction is slightly less than limiting friction and is given by $f_k = \mu_k N$, where $\mu_k < \mu_s$.
- Rolling friction is significantly smaller than sliding friction, which is why wheels are used for transport.
- Angle of friction ($\lambda$) is the angle between the normal reaction and the resultant of limiting friction and normal reaction: $\mu_s = \tan\lambda$.
- Angle of repose ($\alpha$) is the maximum angle of an inclined plane at which a body placed on it just begins to slide: $\mu_s = \tan\alpha$.
- Methods to reduce friction include polishing, lubrication, ball bearings, and streamlining.
Key points
- Friction is a self-adjusting force up to the limiting value.
- Coefficient of friction depends purely on surface pairing and condition.
- Friction is independent of the apparent area of contact within practical limits.
- Work done against friction is dissipated as thermal energy (non-conservative force).
- Walking is impossible without friction.
- Automobiles require friction between tires and road for acceleration, braking, and turning.
Worked examples
- A block of mass $10 \text{ kg}$ rests on a horizontal floor. If $\mu_s = 0.4$ and $\mu_k = 0.3$, find the friction force when a horizontal force of $30 \text{ N}$ is applied ($g = 10 \text{ m/s}^2$). Solution: Normal reaction $N = mg = (10)(10) = 100 \text{ N}$. Max static friction $f_{s,max} = \mu_s N = (0.4)(100) = 40 \text{ N}$. Since the applied force ($30 \text{ N}$) is less than limiting friction ($40 \text{ N}$), the body remains at rest and static friction equals the applied force ($30 \text{ N}$).
- Calculate the angle of repose for a surface with a coefficient of static friction of $0.577$. Solution: $\tan\alpha = 0.577 implies \alpha = \tan^{-1}(0.577) \approx 30^\circ$.
Applications and connections
- Braking systems in automobiles and bicycles.
- Climbing ropes and mountaineering equipment.
- Nails and screws holding structures together.
Advanced points, exceptions and limitations
- At microscopic levels, cold welding of asperities contributes heavily to static friction.
- Stribeck curve explains the transition between boundary lubrication, mixed lubrication, and hydrodynamic lubrication.
- Negative friction phenomena exist in nanoscale atomic force microscopy.
Additional examples
- A car stopping when brakes are applied.
- Writing with a chalk on a black board.
- A block sliding down an inclined plane.
Important distinctions
- Static friction is self-adjusting; kinetic friction is nearly constant.
- Coefficient of static friction is greater than coefficient of kinetic friction ($\mu_s > \mu_k$).
- Sliding friction opposes sliding, whereas rolling friction opposes rolling via deformation.
Common misconceptions
- Believing friction always acts in the direction opposite to motion (it actually opposes *relative* motion and can sometimes drive motion, e.g., walking or car tires pushing ground backward).
- Thinking friction depends on surface area.
- Assuming kinetic friction increases with speed.
Exam focus
- Derivation of acceleration of a body sliding down a rough inclined plane.
- Minimum horizontal force required to move a body on a rough floor.
- Problems involving pulling vs. pushing a lawnmower.
- Angle of repose and angle of friction relationship.
- Power dissipated against frictional forces.
Concept 4 · core
Work, Power, and Energy
Work, energy, and power quantify mechanical energy transformations, conservation laws, and the rate of doing work.
Prerequisites
- Dot product of vectors.
- Newton's laws of motion and kinematic equations.
- Units of measurement in mechanics.
Definitions and scope
- Work ($W$) done by a constant force is defined as the scalar product of force and displacement: $W = \vec{F} \cdot \vec{s} = F s \cos\theta$.
- Kinetic Energy ($KE$) is the energy possessed by a body by virtue of its motion: $KE = \frac{1}{2}mv^2$.
- Potential Energy ($PE$) is the stored energy of a body due to its position, configuration, or state: $PE = mgh$ (gravitational).
- Power ($P$) is the rate at which work is done or energy is transferred: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$.
Step-by-step explanation
- Work can be positive ($0^\circ \leq \theta < 90^\circ$), negative ($90^\circ < \theta \leq 180^\circ$), or zero ($\theta = 90^\circ$).
- Work-Energy Theorem states that the work done by all forces acting on a particle equals the change in its kinetic energy: $W_{net} = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$.
- Conservative forces (e.g., gravitational, electrostatic) are forces where work done is path-independent and depends only on initial and final positions.
- Non-conservative forces (e.g., friction, viscous drag) are path-dependent, and work done along a closed loop is non-zero.
- Potential energy is defined only for conservative forces: $\Delta U = -W_{conservative} = - \int \vec{F} \cdot d\vec{r}$.
- Law of Conservation of Mechanical Energy states that the total mechanical energy ($KE + PE$) of an isolated system remains constant if only conservative forces act.
- Power unit is Watt (W) or Joule per second; commercial unit of electrical energy is kilowatt-hour (kWh), where $1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$.
- Collisions are categorized into perfectly elastic (conservation of both momentum and kinetic energy) and perfectly inelastic (maximum loss of kinetic energy, bodies stick together).
Key points
- Work is a scalar quantity, measured in Joules (J) or Ergs ($1 \text{ J} = 10^7 \text{ ergs}$).
- Centripetal force does zero work because force is perpendicular to displacement ($90^\circ$).
- Potential energy of a spring is given by $U = \frac{1}{2}kx^2$, where $k$ is spring constant.
- Power can also be expressed as the dot product of force and velocity: $P = Fv \cos\theta$.
- Coefficient of restitution ($e$) ranges from $0$ (inelastic) to $1$ (elastic).
- Power output of engines is often measured in horsepower ($1 \text{ HP} = 746 \text{ W}$).
Worked examples
- A force $\vec{F} = (2\hat{i} + 3\hat{j}) \text{ N}$ displaces a body from position $\vec{r}_1 = (1\hat{i} - 1\hat{j}) \text{ m}$ to $\vec{r}_2 = (3\hat{i} + 2\hat{j}) \text{ m}$. Find the work done. Solution: Displacement $\vec{s} = \vec{r}_2 - \vec{r}_1 = (3-1)\hat{i} + (2 - (-1))\hat{j} = 2\hat{i} + 3\hat{j}$. Work $W = \vec{F} \cdot \vec{s} = (2)(2) + (3)(3) = 4 + 9 = 13 \text{ Joules}$.
- An engine pumps $1000 \text{ kg}$ of water per minute from a well $10 \text{ m}$ deep. Calculate the power of the engine ($g = 10 \text{ m/s}^2$). Solution: Work $W = mgh = (1000)(10)(10) = 10^5 \text{ J}$. Time $t = 60 \text{ s}$. Power $P = \frac{W}{t} = \frac{10^5}{60} = 1666.67 \text{ W}$.
Applications and connections
- Hydroelectric power generation utilizing potential energy of stored water.
- Ballistic pendulum experiments to measure bullet velocities.
- Designing shock absorbers and spring systems in vehicles.
Advanced points, exceptions and limitations
- Potential energy curves ($U$ vs $x$) provide stability analysis: minima represent stable equilibrium, maxima represent unstable equilibrium, and points of inflection represent neutral equilibrium.
- Force is related to potential energy gradient by $F_x = -\frac{dU}{dx}$.
- In inelastic collisions, kinetic energy lost appears as heat, sound, or permanent deformation.
Additional examples
- Lifting a suitcase vertically upward against gravity (positive work).
- Car brakes bringing a vehicle to a halt (negative work).
- A coolie carrying luggage on a flat platform (zero work against gravity).
Important distinctions
- Conservative forces conserve mechanical energy; non-conservative forces dissipate it into thermal energy.
- Kinetic energy depends on speed (frame-dependent); potential energy depends on configuration/position.
- Work is energy in transit; energy is stored capacity to do work.
Common misconceptions
- Thinking work is done whenever a force is applied, ignoring displacement.
- Believing total energy of the universe is conserved *and* mechanical energy is always conserved (mechanical energy is only conserved in conservative systems).
- Confusing power with energy.
Exam focus
- Numerical problems using Work-Energy theorem.
- Potential energy function and finding force using derivative $F = -dU/dx$.
- Elastic collision formulas in 1D (velocities after collision).
- Power calculations involving pumping water or elevators.
- Equilibrium points from potential energy curves.
Concept 5 · advanced
Gravitation and Planetary Motion
Universal gravitation explains celestial mechanics, satellite motion, and variation of gravity across Earth.
Prerequisites
- Newton's laws of motion.
- Circular motion and centripetal acceleration.
- Basic calculus and algebra.
Definitions and scope
- Newton's Law of Gravitation states that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: $F = G \frac{m_1 m_2}{r^2}$.
- Universal Gravitational Constant ($G$) is $6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$.
- Acceleration due to gravity ($g$) is the acceleration experienced by a body in free fall under Earth's gravitational pull.
- Escape velocity ($v_e$) is the minimum velocity required for an object to break free from Earth's gravitational field completely.
Step-by-step explanation
- Gravitational force is a central, conservative force acting along the line joining the centers of two masses.
- Variation of $g$ with altitude: $g' = g\left(1 - \frac{2h}{R}\right)$ for small heights, and $g' = g \frac{R^2}{(R+h)^2}$ generally.
- Variation of $g$ with depth: $g' = g\left(1 - \frac{d}{R}\right)$, showing gravity is zero at Earth's center.
- Variation of $g$ with latitude: $g$ is maximum at poles and minimum at the equator due to Earth's rotation and equatorial bulge.
- Kepler's First Law (Law of Orbits): All planets move in elliptical orbits with the Sun at one of the foci.
- Kepler's Second Law (Law of Areas): A planet-Sun line sweeps out equal areas in equal intervals of time, implying conservation of angular momentum.
- Kepler's Third Law (Law of Periods): The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of its orbit: $T^2 \propto r^3$.
- Orbital velocity of a satellite close to Earth: $v_o = \sqrt{gR} \approx 7.9 \text{ km/s}$; Escape velocity: $v_e = \sqrt{2gR} = \sqrt{2} v_o \approx 11.2 \text{ km/s}$.
Key points
- Gravitational force is the weakest of the four fundamental forces in nature, yet dominant on a cosmic scale.
- Gravitational potential energy at distance $r$ from Earth is $U = -\frac{GMm}{r}$.
- Binding energy of a satellite is equal to its positive orbital energy magnitude.
- Weightlessness in an orbiting satellite arises because the satellite and astronaut are in free fall together toward Earth.
- Geostationary satellites have a time period of $24 \text{ hours}$ and orbit at an altitude of $\approx 35,786 \text{ km}$ above the equator.
- Polar satellites orbit from pole to pole at lower altitudes for remote sensing and weather monitoring.
Worked examples
- Calculate the escape velocity from Earth given $g = 9.8 \text{ m/s}^2$ and Earth's radius $R = 6400 \text{ km}$. Solution: $v_e = \sqrt{2gR} = \sqrt{2 \times 9.8 \times 6.4 \times 10^6} = \sqrt{1.2544 \times 10^8} = 11,200 \text{ m/s} = 11.2 \text{ km/s}$.
- If the radius of Earth were to shrink by $1\%$ with its mass remaining constant, find the approximate percentage change in acceleration due to gravity $g$. Solution: $g = \frac{GM}{R^2} \implies \frac{\Delta g}{g} = -2 \frac{\Delta R}{R}$. Since $\frac{\Delta R}{R} = -1\%$, $\frac{\Delta g}{g} = -2(-1\%) = +2\%$. Thus, $g$ increases by $2\%$.
Applications and connections
- Launching and positioning communication and meteorological satellites.
- GPS navigation systems relying on precise satellite orbital mechanics.
- Predicting planetary positions, eclipses, and cometary returns.
Advanced points, exceptions and limitations
- Gravitational self-energy of a uniform solid sphere of mass $M$ and radius $R$ is $U = -\frac{3GM^5}{5R}$.
- Tidal forces arise due to the differential gravitational pull across an extended body.
- General relativity generalizes gravitation as spacetime curvature rather than a classical force field.
Additional examples
- Moon orbiting around Earth.
- An apple falling from a tree to the ground.
- Artificial satellites transmitting television signals.
Important distinctions
- Orbital velocity is $\sqrt{GM/R}$, while escape velocity is $\sqrt{2GM/R}$ (factor of $\sqrt{2}$).
- Geostationary satellites rotate synchronously with Earth; polar satellites orbit perpendicularly to the equator.
- Gravitational force is always attractive, unlike electrostatic or magnetic forces which can be repulsive.
Common misconceptions
- Thinking astronauts in space stations float because there is zero gravity (gravity is actually about $90\%$ of surface value at ISS altitude; they float because of free-fall orbital motion).
- Believing Kepler's orbits are highly eccentric circles (most planets have nearly circular orbits).
- Assuming $g$ is constant everywhere on Earth.
Exam focus
- Derivation of Kepler's Third Law from Newton's law of gravitation.
- Numerical problems on variation of $g$ with height and depth.
- Escape velocity and total mechanical energy of satellites.
- Time period of a simple pendulum inside an orbiting satellite (zero effective $g$, so infinite time period).
- Comparison of geostationary and polar orbit characteristics.
Concept 6 · advanced
Circular Motion and Rotational Dynamics
Circular motion involves objects moving along a curved circular path, requiring centripetal force and rotational analogs.
Prerequisites
- Vectors and trigonometric functions.
- Newton's laws of motion.
- Work and kinetic energy.
Definitions and scope
- Angular displacement ($\theta$) is the angle swept out by the radius vector in radians.
- Angular velocity ($\omega$) is the rate of change of angular displacement: $\omega = \frac{d\theta}{dt}$.
- Centripetal acceleration ($a_c$) is the acceleration directed towards the center of the circular path: $a_c = \frac{v^2}{r} = \omega^2 r$.
- Moment of inertia ($I$) is the rotational inertia of a body, given by $I = \sum m_i r_i^2$.
Step-by-step explanation
- Relation between linear and angular velocity: $v = r\omega$.
- Centripetal force is not a new fundamental force; it is the net inward force (tension, gravity, friction, or normal force) that enables circular motion: $F_c = \frac{mv^2}{r} = m\omega^2 r$.
- Centrifugal force is a pseudo force experienced in a rotating non-inertial frame of reference, equal in magnitude ($m\omega^2 r$) but opposite in direction to centripetal force.
- Banking of roads reduces reliance on tire friction by tilting the road surface, providing the necessary centripetal component: $\tan\theta = \frac{v^2}{rg}$.
- Torque ($\tau$) is the rotational analog of force, defined as $\vec{\tau} = \vec{r} \times \vec{F}$, with magnitude $\tau = r F \sin\phi$.
- Rotational kinetic energy is given by $KE_{rot} = \frac{1}{2}I\omega^2$.
- Angular momentum ($L$) is defined as $\vec{L} = \vec{r} \times \vec{p} = I\omega$, and is conserved in the absence of external torque.
- Parallel axis theorem ($I = I_{cm} + md^2$) and perpendicular axis theorem ($I_z = I_x + I_y$ for planar bodies) facilitate calculating moments of inertia.
Key points
- Uniform circular motion has constant speed but changing velocity vector due to continuous change in direction.
- Tangential acceleration changes speed in non-uniform circular motion, while centripetal acceleration changes direction.
- Unit of angular velocity is rad/s; unit of torque is $\text{N}\cdot\text{m}$.
- Radius of gyration ($k$) relates moment of inertia and mass by $I = mk^2$.
- Rolling motion without slipping combines translational motion of the center of mass with rotational motion about the center of mass.
- A rigid body in equilibrium requires both net force to be zero (translational) and net torque to be zero (rotational).
Worked examples
- A stone of mass $0.5 \text{ kg}$ is tied to a string of length $1.5 \text{ m}$ and whirled in a horizontal circle at a constant speed of $6 \text{ m/s}$. Find the tension in the string. Solution: $m = 0.5$, $r = 1.5$, $v = 6$. Tension provides centripetal force: $T = \frac{mv^2}{r} = \frac{(0.5)(6^2)}{1.5} = \frac{(0.5)(36)}{1.5} = 12 \text{ N}$.
- Calculate the maximum safe speed for a car to negotiate a curved road of radius $100 \text{ m}$ with a coefficient of friction of $0.4$ ($g = 10 \text{ m/s}^2$). Solution: $v_{\max} = \sqrt{\mu rg} = \sqrt{(0.4)(100)(10)} = \sqrt{400} = 20 \text{ m/s}$ ($72 \text{ km/h}$).
Applications and connections
- Banking of highway curves and railway tracks to prevent skidding.
- Centrifuges used in medical laboratories for blood separation.
- Ice skaters spinning faster by pulling their arms inward (conservation of angular momentum).
Advanced points, exceptions and limitations
- Coriolis force is another pseudo force in rotating reference systems that deflects moving objects perpendicular to their velocity vector.
- Precession occurs when a spinning top or gyroscope experiences an external torque perpendicular to its angular momentum.
- Total kinetic energy of a rolling body is the sum of translational and rotational kinetic energies: $KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = \frac{1}{2}mv^2\left(1 + \frac{k^2}{R^2}\right)$.
Additional examples
- A car turning around a circular track.
- Earth rotating around the Sun.
- A giant wheel in an amusement park.
Important distinctions
- Centripetal force acts towards the center in inertial frames; centrifugal force acts outwards in non-inertial rotating frames.
- Mass is translational inertia; moment of inertia is rotational inertia.
- Force causes linear acceleration; torque causes angular acceleration.
Common misconceptions
- Believing centrifugal force pulls objects outward in an inertial frame (centrifugal force only exists in rotating frames).
- Thinking uniform circular motion has zero acceleration (it has centripetal acceleration $v^2/r$).
- Confusing torque with work.
Exam focus
- Derivation of maximum safe speed on a banked road with friction.
- Moments of inertia for standard symmetric bodies (ring, disc, sphere, cylinder, rod).
- Conservation of angular momentum problems (e.g., skater pulling arms in, star collapsing).
- Rolling down an inclined plane (acceleration and velocity formulas).
- Parallel and perpendicular axis theorem applications.