Plus One

Chapter 1: Physical World
Concepts and Terms

• Physics: Systematic study of nature through observation, pattern-finding, and modelling; aims to explain diverse phenomena using a small set of concepts and universal laws.
• Scope of Physics:
  • Macroscopic domain: Lab, terrestrial, astronomical scales (classical physics).
  • Microscopic domain: Atomic, molecular, nuclear scales (quantum theory).
  • Mesoscopic: Between micro and macro (few tens/hundreds of atoms).
• Classical sub-disciplines:
  • Mechanics, Electrodynamics, Optics, Thermodynamics.
• Fundamental forces:
  • Gravitational: Universal, always attractive, infinite range.
  • Electromagnetic: Acts between charges/magnets, attractive or repulsive, infinite range.
  • Strong nuclear: Among nucleons/hadrons, very short range (~10^-15 m).
  • Weak nuclear: In certain particle interactions (e.g., beta decay), very short range (~10^-16 m).
• Nature of physical laws:
  • Quantitative, predictive, testable.
  • Built on hypotheses/axioms/models.
  • Universal applicability.
  • Conserved quantities: Energy, linear momentum, angular momentum, charge; closely tied to symmetries of nature.

Exam-Oriented Q&A

1. Define physics and its goal.

• Physics seeks to understand natural phenomena via careful observation, pattern recognition, and models, aiming to explain them with universal laws.

2. Macroscopic vs microscopic domains (with examples).

• Macroscopic: Rockets, planetary motion, fluids in labs. Microscopic: Atoms, nuclei, electrons, photons. Classical vs quantum descriptions respectively.

3. List the four fundamental forces with a key trait each.

• Gravitational (always attractive), Electromagnetic (attractive/repulsive), Strong (short-range, strongest at nuclear scale), Weak (very short-range, responsible for certain decays).

4. What are conserved quantities? Why important?

• Quantities unchanged in time for isolated systems (energy, momentum, angular momentum, charge). They constrain and predict physical processes; linked to symmetries.

5. Role of hypotheses. Can universal laws be “proved”?

• Hypotheses guide explanations and predictions; universal laws are not proved absolutely—only supported or refuted by experiments.

Chapter 2: Units and Measurement
Concepts and Terms

• Measurement: Comparing a quantity with a standard unit.
• Units:
  • Base (fundamental): m, kg, s, A, K, mol, cd.
  • Derived: Combinations of base units.
• Systems of units: CGS, FPS, MKS, SI (decimal-based).
• SI notes: Seven base units; radian and steradian are dimensionless derived units (for plane and solid angles).
• Accuracy vs precision: Closeness to true value vs resolution/repeatability.
• Errors: Absolute, mean absolute, relative, percentage; significant figures indicate precision.
• Dimensions: Express derived quantities via powers of base quantities (e.g., Force: [M L T^-2]).
• Dimensional analysis: Check homogeneity, derive relations (up to constants), estimate dependencies; cannot yield dimensionless factors.

Equations

• Mean: ā = (a1 + a2 + … + an)/n
• Absolute error: Δai = |ai − ā|; Mean absolute error: (Σ|Δai|)/n
• Relative error: (Δa_mean)/ā; Percentage error: 100 × (Δa_mean)/ā
• Error rules:
  • Sum/difference: ΔZ = ΔA + ΔB
  • Product/quotient: (ΔZ/Z) = (ΔA/A) + (ΔB/B)
  • Powers: (ΔZ/Z) = p(ΔA/A) + q(ΔB/B) + r(ΔC/C) for Z = A^p B^q/C^r
• Example dimensions: Volume [L^3], Velocity [L T^-1], Acceleration [L T^-2], Density [M L^-3], Force [M L T^-2].

Exam-Oriented Q&A

1. Accuracy vs precision.

• Accuracy: Closeness to true value. Precision: Fineness/repeatability of measurement.

2. Error in sum/difference.

• ΔZ = ΔA + ΔB for Z = A ± B.

3. Error in product/quotient.

• (ΔZ/Z) = (ΔA/A) + (ΔB/B) for Z = AB or A/B.

4. Dimensions and dimensional formula for force.

• Dimensions are exponents of base quantities. Force: [M L T^-2].

5. Dimensional consistency check.

• All terms must share the same dimensions; inconsistency implies the equation is wrong.

Chapter 3: Motion in a Straight Line
Concepts and Terms

• Motion: Change in position with time.
• Frame of reference: Coordinate system plus clock.
• Point object: Size negligible compared to path scale.
• Path length (scalar) vs displacement (vector).
• Average velocity: Δx/Δt; Average speed: total path/total time.
• Instantaneous velocity: v = dx/dt; slope of x–t tangent.
• Acceleration: a = dv/dt; slope of v–t tangent.
• Uniform acceleration kinematics.
• Free fall (ignore air resistance).
• Stopping distance ∝ (initial speed)^2.
• Reaction time: Perception-to-action delay.
• Relative velocity: vAB = vA − vB in 1D.

Equations

• Δx = x2 − x1
• v_avg = Δx/Δt; speed_avg = total path/total time
• v = dx/dt; a = dv/dt
• For constant a (with x0 = 0 at t = 0):
  • v = v0 + at
  • x = v0 t + (1/2) a t^2
  • v^2 = v0^2 + 2 a x
• Stopping distance (magnitude): ds = v0^2/(2|a|)
• Relative velocity (1D): vAB = vA − vB

Exam-Oriented Q&A

1. Path length vs displacement with example.

• Path length counts total ground covered; displacement is straight-line change in position with direction. Out-and-back motion can have large path length but small displacement.

2. Average vs instantaneous velocity; x–t graph meaning.

• Average: total displacement/time; slope of secant. Instantaneous: dx/dt; slope of tangent.

3. Three kinematic equations (constant a).

• v = v0 + at; x = v0 t + (1/2) a t^2; v^2 = v0^2 + 2 a x.

4. Stopping distance dependence on speed.

• ds ∝ v0^2; doubling speed quadruples stopping distance (same deceleration).

5. Relative velocity in 1D.

• Velocity of B relative to A: vBA = vB − vA.

Chapter 4: Motion in a Plane
Concepts and Terms

• Scalars vs vectors: Magnitude only vs magnitude + direction.
• Position vector r; displacement Δr = r' − r (path independent).
• Equality of vectors: Same magnitude and direction.
• Multiply vector by scalar: Changes magnitude; sign flips direction.
• Vector addition: Head-to-tail (triangle) or parallelogram; commutative and associative; subtraction as A + (−B).
• Unit vectors î, ĵ, k̂; components Ax = A cosθ, Ay = A sinθ; A = Ax î + Ay ĵ.
• Analytical addition: Rx = Ax + Bx, Ry = Ay + By.
• Velocity and acceleration in 2D: v = dr/dt, a = dv/dt; components along axes; v tangent to path.
• Constant-acceleration motion in 2D = two independent 1D motions.
• Projectile motion: Horizontal uniform motion + vertical uniformly accelerated motion; trajectory is a parabola.
• Uniform circular motion: Speed constant; acceleration is centripetal, toward center.

Equations

• Vector components: A = Ax î + Ay ĵ; |A| = √(Ax^2 + Ay^2); tanθ = Ay/Ax
• Resultant magnitude (law of cosines): R = √(A^2 + B^2 + 2AB cosθ)
• r = x î + y ĵ; Δr = Δx î + Δy ĵ
• vx = dx/dt; vy = dy/dt; ax = dvx/dt; ay = dvy/dt
• Constant a (vector form): v = v0 + a t; r = r0 + v0 t + (1/2) a t^2
• Projectile (origin launch, no air drag):
  • x = (v0 cosθ0) t; y = (v0 sinθ0) t − (1/2) g t^2
  • vx = v0 cosθ0; vy = v0 sinθ0 − g t
  • Path: y = (tanθ0) x − (g x^2)/(2 v0^2 cos^2θ0)
  • hmax = (v0^2 sin^2θ0)/(2g); t_to_hmax = (v0 sinθ0)/g
  • Range: R = (v0^2 sin2θ0)/g; Rmax at 45°: v0^2/g
• Uniform circular motion:
  • v = R ω; ac = v^2/R = ω^2 R
  • T = 2π/ω; ν = 1/T; ac = 4π^2 ν^2 R

Exam-Oriented Q&A

1. Define scalar and vector with two examples.

• Scalars: mass, temperature. Vectors: displacement, force.

2. Head-to-tail addition and commutativity.

• Place tail of B at head of A; resultant from tail of A to head of B. A + B = B + A.

3. Unit vectors and resolving a vector.

• Unit vectors specify directions along axes; A = Ax î + Ay ĵ (+ Az k̂ in 3D).

4. Projectile range and maximum height.

• R = (v0^2 sin2θ0)/g; hmax = (v0^2 sin^2θ0)/(2g).

5. Meaning of “uniform” in UCM and direction of acceleration.

• Uniform = constant speed. Acceleration is centripetal, toward center.

Chapter 5: Laws of Motion
Concepts and Terms

• Aristotle’s fallacy: Belief that force is needed to maintain motion.
• Newton’s First Law (inertia): Body stays at rest or uniform straight-line motion unless acted on by net external force.
• Momentum p = m v (vector).
• Newton’s Second Law: F = dp/dt; for constant mass, F = m a (vector).
• Impulse: J = F Δt = Δp; impulsive forces act over short time with finite Δp.
• Newton’s Third Law: Forces come in equal and opposite pairs on different bodies.
• Conservation of momentum: Total momentum constant if external force is zero.
• Equilibrium of a particle: Net force zero → zero acceleration.
• Common forces: Gravity, normal, tension, spring, friction (static fs ≤ μs N; kinetic fk = μk N with μk < μs).
• Circular motion dynamics: Centripetal force requirement mv^2/R provided by existing forces; banking reduces reliance on friction.

Equations

• p = m v; F = dp/dt = m a
• Impulse J = Δp
• Action–reaction: FAB = −FBA
• Momentum conservation: If Fext = 0, total P is constant
• Friction: fs ≤ μs N; fk = μk N
• Centripetal acceleration: ac = v^2/R; Centripetal force: Fc = m v^2/R

Exam-Oriented Q&A

1. State First Law; implication for equilibrium.

• Net external force zero → zero acceleration → rest or uniform velocity.

2. Define momentum; Second Law in momentum form and link to F = ma.

• F = dp/dt; with constant mass, dp/dt = m a ⇒ F = m a.

3. Define impulse with example.

• Impulse = change in momentum (e.g., bat hitting a ball over a short contact time).

4. Explain Third Law and where forces act.

• Equal and opposite forces act on different bodies; they don’t cancel on the same body.

5. State momentum conservation and its condition.

• Total momentum conserved for an isolated system (no net external force).

Chapter 6: Work, Energy and Power
Concepts and Terms

• Dot product: A·B = AB cosθ (scalar); commutative, distributive.
• Work: W = F d cosθ = F·d; positive/negative/zero depending on angle and displacement.
• Kinetic energy: K = (1/2) m v^2.
• Work–Energy Theorem: ΔK = W_net.
• Conservative forces: Path-independent work; zero work over closed loops; F = −dV/dx; examples: gravity, spring.
• Potential energy: Stored energy by configuration/position (gravity: mgh near Earth; general gravity: −Gm1m2/r; spring: (1/2) k x^2).
• Conservation of mechanical energy: K + V = constant (only conservative forces doing work).
• Forms of energy: Mechanical, thermal, chemical, electrical, nuclear, etc.
• Mass–energy equivalence: E = m c^2.
• Conservation of energy: Total energy of an isolated system is constant.
• Power: Rate of doing work/energy transfer; P_avg = W/t; instantaneous P = dW/dt = F·v; units: W (J/s); 1 kWh = 3.6×10^6 J.
• Collisions:
  • Momentum conserved in all collisions.
  • Elastic: Kinetic energy conserved.
  • Inelastic: Kinetic energy not conserved; completely inelastic → objects stick.

Equations

• Work: W = F d cosθ
• K = (1/2) m v^2; ΔK = W_net
• Variable force (1D): W = ∫ F(x) dx (from xi to xf)
• Conservative force: F(x) = −dV/dx
• Vgravity near surface: mgh; general: −G m1m2/r
• Vspring: (1/2) k x^2
• Power: P = dW/dt = F·v; 1 kWh = 3.6×10^6 J
• 1D collisions:
  • Completely inelastic (stick): m1 v1i + m2 v2i = (m1 + m2) vf
  • Elastic (1D): Momentum and kinetic energy both conserved

Exam-Oriented Q&A

1. Define work; when is it zero?

• W = F·d. Zero if displacement is zero, force is zero, or force ⟂ displacement.

2. State the Work–Energy Theorem and what info it omits vs F = ma.

• ΔK = W_net. It loses time and directional details present in the vector form of Newton’s second law.

3. Define conservative force; examples; potential energy linkage.

• Path-independent work; gravity/spring; F = −dV/dx with a potential energy function V.

4. Define power and SI unit; kW vs kWh.

• Power = rate of work; unit: watt (W). kW is power; kWh is energy.

5. Elastic vs inelastic collisions.

• Both conserve momentum; elastic also conserves kinetic energy; inelastic does not.

Chapter 7: Systems of Particles and Rotational Motion
Concepts and Terms

• Rigid body: Distances between constituent particles fixed.
• Pure translation: All points have same velocity.
• Pure rotation: About a fixed axis; each point moves in a circle centered on the axis.
• Rolling: Translation + rotation.
• Centre of mass (CM): Mass-weighted average position; external forces determine CM motion.
• Velocity of CM: P = M V (total momentum equals total mass times CM velocity).
• Second law for a system: F_ext = dP/dt.
• Cross product: a × b, magnitude ab sinθ, direction by right-hand rule, not commutative.
• Angular velocity ω = dθ/dt (vector along axis); angular acceleration α = dω/dt.
• Torque τ = r × F.
• Angular momentum L = r × p; for system, L_total = Σ (ri × pi).
• τ_ext = dL/dt.
• Conservation of angular momentum: If τ_ext = 0, L is constant; for fixed axis, I ω = constant.
• Equilibrium of a rigid body: ΣF_ext = 0 and Στ_ext = 0.
• Centre of gravity: Point where total gravitational torque is zero (coincides with CM in uniform g).
• Moment of inertia I = Σ mi ri^2; depends on mass distribution and axis.
• Theorems:
  • Perpendicular axes (lamina): Iz = Ix + Iy.
  • Parallel axes: I' = Icm + M a^2.
• Rotational dynamics: τ = I α (analogue of F = m a).
• Rotational kinetic energy: K = (1/2) I ω^2; Power: P = τ ω.
• Rolling without slipping: vCM = R ω; K_total = (1/2) M vCM^2 + (1/2) I ω^2.

Equations

• CM (discrete): R = (Σ mi ri)/M; V = P/M
• v (point in rigid rotation): v = ω × r
• τ = r × F; L = r × p; dL/dt = τ_ext
• Equilibrium: ΣF = 0; Στ = 0
• I = Σ mi ri^2
• Krot = (1/2) I ω^2; P = τ ω
• Perpendicular axes: Iz = Ix + Iy (planar body)
• Parallel axes: I' = Icm + M a^2
• Rotational kinematics (constant α):
  • ω = ω0 + α t
  • θ = θ0 + ω0 t + (1/2) α t^2
  • ω^2 = ω0^2 + 2 α (θ − θ0)

Exam-Oriented Q&A

1. Define CM and its significance.

• R = (Σ mi ri)/M. External forces move the CM as if all mass were concentrated there.

2. Conditions for mechanical equilibrium.

• ΣF_ext = 0 (no CM acceleration) and Στ_ext = 0 (no angular acceleration).

3. Moment of inertia and why it’s rotational analogue of mass.

• I measures resistance to change in rotational motion; larger I → harder to change ω.

4. State the parallel-axes theorem.

• I' = Icm + M a^2, where a is distance between axes.

5. Conservation of angular momentum about a fixed axis.

• If τ_ext = 0 about that axis, Lz = I ω is constant.

Chapter 8: Gravitation
Concepts and Terms

• Newton’s law of universal gravitation: F = G m1 m2 / r^2; attractive; acts along the line joining masses.
• Gravitational constant G ≈ 6.67 × 10^-11 N m^2 kg^-2.
• Superposition: Net gravitational force is vector sum of individual forces.
• Kepler’s laws:
  • Orbits: Elliptical, Sun at a focus.
  • Areas: Equal areas in equal times (consequence of angular momentum conservation).
  • Periods: T^2 ∝ R^3 (R is semi-major axis).
• Central forces: Directed along r; magnitude depends only on r; conserve angular momentum; motion planar.
• Acceleration due to gravity g:
  • At Earth’s surface: g = G ME / RE^2.
  • At altitude h: g(h) = G ME / (RE + h)^2 (decreases with h).
  • At depth d (uniform density): g(d) = g (1 − d/RE) (decreases with d).
• Gravitational potential energy: V = −G m1 m2 / r; zero at r → ∞; negative for bound systems.
• Escape speed: Minimum speed to reach infinity with zero KE (no air drag).
• Earth satellites:
  • Orbital speed (circular): v = √(G ME / (RE + h)).
  • Period: T^2 = (4π^2/(G ME)) (RE + h)^3.
  • Total energy: E = −G ME m / (2 (RE + h)) (negative: bound).
• Geostationary satellite: Equatorial orbit, period ≈ 24 h, appears stationary.
• Polar satellite: N–S orbit over poles, useful for mapping/weather.
• Weightlessness: In orbit, astronaut and craft are in free fall together → no normal force → apparent weightlessness.

Equations

• F = G m1 m2 / r^2
• g = G ME / RE^2; g(h) = G ME / (RE + h)^2; g(d) = g (1 − d/RE)
• V = −G m1 m2 / r
• Escape speed: ve = √(2 G ME / RE) = √(2 g RE)
• Orbital speed: v = √(G ME / (RE + h))
• Kepler (Earth satellite form): T^2 = (4π^2/(G ME)) (RE + h)^3
• Satellite total energy: E = −G ME m / (2 (RE + h))

Exam-Oriented Q&A

1. State Newton’s law of gravitation and role of G.

• Force ∝ m1 m2 / r^2; G makes it an equality and sets the strength of gravity universally.

2. Kepler’s three laws.

• Elliptical orbits; equal areas in equal times; T^2 ∝ R^3.

3. Variation of g with altitude and depth.

• Decreases with both altitude (∝ 1/(RE + h)^2) and depth (linearly under uniform density).

4. Define gravitational potential energy; why is satellite energy negative?

• V = −G m1 m2 / r with zero at infinity; bound orbits have negative total energy.

5. What is escape speed? Why weightlessness in orbit?

• Minimum speed to reach infinity with zero KE. In orbit, continuous free fall eliminates normal force, so apparent weight is zero.