Core Formula/Logic: Time period of a simple pendulum T = 2π sqrt(L/g), where L is length and g is gravitational acceleration.
Step-by-Step Solution: 1. On Earth: T_earth = 2π sqrt(L/g_earth). 2. On Moon: g_moon = g_earth/6. 3. T_moon = 2π sqrt(L/(g_earth/6)) = 2π sqrt(6L/g_earth) = sqrt(6) × 2π sqrt(L/g_earth) = sqrt(6) × T_earth. 4. Since T_moon > T_earth, the clock runs slower by factor sqrt(6).
Common Pitfall: Forgetting the square root gives T_moon = 6 × T_earth (option B). Confusing faster/slower direction gives sqrt(6) times faster (option C).
Shortcut/Takeaway: Period ∝ 1/sqrt(g). If g decreases to 1/6th, period increases by sqrt(6). Clock runs slower when period increases.

Core Formula/Logic: Spring constant k = F/x, where F is force and x is extension. For springs of same material and cross-section, k ∝ 1/L (inversely proportional to length).
Step-by-Step Solution: 1. Original spring: length L, spring constant k. 2. Each half has length L/2. 3. Since k ∝ 1/L, new constant k' = k × (L/(L/2)) = 2k. 4. Alternatively, two identical springs of constant k' in series: 1/k = 1/k' + 1/k' = 2/k' → k' = 2k.
Common Pitfall: Assuming constant remains k (option A) ignores length dependence. Thinking k halves (option B) reverses proportionality. Squaring factor (option D) incorrectly uses k ∝ 1/L^2.
Shortcut/Takeaway: Cutting a spring into n equal parts multiplies spring constant by n. Halving length doubles k.

Core Formula/Logic: Vector quantities possess both magnitude and direction, while scalars have only magnitude. Angular momentum L = r x p (cross product of position and linear momentum vectors).
Step-by-Step Solution: 1. Heat measures thermal energy transfer - scalar (no direction).
2. Angular momentum has direction perpendicular to rotation plane - vector.
3. Time measures duration - scalar.
4. Work = force·displacement·cosθ - scalar (dot product).
Common Pitfall: Confusing work (scalar dot product) with force (vector) produces option D. Misremembering angular momentum as scalar leads to missing option B.
Shortcut/Takeaway: Cross products always yield vectors. Common vectors: displacement, velocity, force, momentum, torque, angular momentum. Common scalars: mass, time, temperature, energy, work, power.

Why Correct: The pilot shares the same horizontal velocity as the bomb at release, so relative to the pilot, the bomb has zero horizontal velocity and appears to fall straight down vertically.
Distractor Analysis: A curved path with fall behind occurs when observed from a stationary ground observer who sees the bomb's horizontal motion. A curved path with fall ahead would require the bomb to have greater horizontal velocity than the plane. Remaining stationary violates gravity's constant downward acceleration of 9.8 m/s².
Takeaway: For any object dropped from a moving platform, observers moving with the platform see only vertical motion, while stationary observers see projectile motion with both horizontal and vertical components.

Core Formula/Logic: Newton's first law of motion: an object in motion continues in a straight line at constant velocity unless acted upon by a net external force. Centripetal force acts radially inward to maintain circular motion.
Step-by-Step Solution: 1. While whirling, centripetal force (tension in rope) pulls stone inward, changing direction continuously.
2. When rope snaps, centripetal force becomes zero.
3. Stone's inertia causes it to continue moving in the direction of its instantaneous velocity at the moment of snapping.
4. In circular motion, instantaneous velocity is tangential to the circle.
Common Pitfall: Thinking centripetal force pushes outward (centrifugal misconception) leads to selecting radially outward. Confusing centripetal direction leads to radially inward.
Shortcut/Takeaway: Remember: "No centripetal force → motion becomes tangential." This applies to any circular motion when centripetal force is removed (e.g., car skidding on curve, satellite engine failure).

Why Correct: Two unequal forces produce a net force causing translational motion, while their offset lines of action create a torque causing rotational motion.
Distractor Analysis: Only rotational motion occurs when equal and opposite forces form a pure couple. Only translational motion requires forces along the same line of action. Neither motion requires balanced forces with zero net force and zero torque.
Takeaway: A pure couple (equal, opposite, parallel forces) produces only rotation without translation.

Why Correct: Frictional force is non-conservative — work done depends on path taken, mechanical energy converts to heat, and total mechanical energy is not conserved.
Distractor Analysis: Gravitational force is conservative — work done is path-independent and depends only on vertical displacement. Electrostatic force between stationary charges is conservative — work done moving a charge depends only on endpoints, not path. Magneto static force (magnetic force on moving charges) is also conservative in static magnetic fields — work done is zero since force is perpendicular to velocity.
Takeaway: Conservative forces have zero curl (∇ × F = 0) and can be expressed as gradient of a potential function (F = -∇V).

Why Correct: Dyne-sec is the CGS unit of momentum, where dyne is the unit of force and second is the unit of time, making it equivalent to g·cm/s.
Distractor Analysis: Force uses dyne alone, energy uses erg, and power uses erg/sec.
Takeaway: In SI units, momentum is measured in kg·m/s, which equals Newton-second (N·s).

Core Formula/Logic: Momentum p = sqrt(2mK) for kinetic energy K, so for fixed K, momentum increases with mass.
Step-by-Step Solution: 1. Kinetic energy K = p^2/(2m). 2. Rearranging gives p = sqrt(2mK). 3. For constant K, p ∝ sqrt(m). 4. Masses: electron ≈ 9.1e-31 kg, proton ≈ 1.67e-27 kg, deuteron ≈ 3.34e-27 kg (proton+neutron), alpha ≈ 6.64e-27 kg (2 protons+2 neutrons). 5. Alpha particle has highest mass, so highest momentum.
Common Pitfall: Assuming lighter particles have higher momentum due to higher speed at same K, but p depends on sqrt(m), not m directly, giving alpha highest p.
Shortcut/Takeaway: For same K, p ∝ sqrt(m); compare masses: alpha > deuteron > proton > electron.

Core Formula/Logic: Acceleration is defined as the rate of change of velocity (a = Δv/Δt). Constant speed means velocity magnitude doesn't change, and horizontal surface means direction doesn't change, so velocity is constant and acceleration is zero.
Step-by-Step Solution: 1. Constant speed means magnitude of velocity is unchanged. 2. Horizontal surface means direction is unchanged (straight line). 3. Constant velocity (both magnitude and direction) means Δv = 0. 4. Acceleration a = Δv/Δt = 0/Δt = 0.
Common Pitfall: Confusing speed (scalar) with velocity (vector) leads to thinking velocity is zero. Confusing momentum (p = mv) with acceleration leads to selecting momentum instead.
Shortcut/Takeaway: "Constant speed + straight line = zero acceleration" is an absolute rule. Velocity, momentum, and kinetic energy (KE = 1/2 mv2) all exist with constant speed.

Core Formula/Logic: Scalar quantities have only magnitude and no direction, while vector quantities have both magnitude and direction.
Step-by-Step Solution: 1. Velocity is a vector (speed with direction).
2. Force is a vector (has magnitude and direction).
3. Momentum is a vector (mass × velocity).
4. Energy is a scalar (e.g., kinetic energy = 1/2 × mass × speed2 has no direction).
Common Pitfall: Confusing momentum (vector) with energy (scalar) because both involve motion, but momentum includes direction while energy does not.
Shortcut/Takeaway: Common scalars: mass, distance, speed, energy, temperature, time. Common vectors: displacement, velocity, acceleration, force, momentum.

Why Correct: Newton's first law defines inertia, stating an object remains at rest or in uniform motion unless acted upon by a net external force, but it does not provide the quantitative definition of force.
Distractor Analysis: True incorrectly suggests the first law defines force, but it only describes the effect of force on motion without defining force itself. Newton's first law defines inertia, not force defines inertia as the tendency of objects to resist changes in motion. Newton's second law defines force mathematically defines force as the product of mass and acceleration.
Takeaway: Newton's second law (F = ma) provides the operational definition of force as the product of mass and acceleration.

Core Formula/Logic: Impulse-momentum theorem: Force × time = change in momentum (FΔt = Δp).
Step-by-Step Solution: 1. For body A: Impulse = F × t = ΔpA.
2. For body B: Impulse = F × t = ΔpB.
3. Since F and t are identical for both, ΔpA = ΔpB.
4. Assuming both start from rest, final momentum pA = pB.
Common Pitfall: Confusing momentum with velocity leads to option A. Assuming heavier mass means greater momentum leads to option D. Using F=ma alone without considering time leads to incorrect velocity comparisons.
Shortcut/Takeaway: For same force and same time interval, impulse (FΔt) is identical → change in momentum is identical regardless of mass.

Why Correct: Newton's second law states F = ma, where force equals mass times acceleration. Stopping a vehicle involves deceleration (negative acceleration), and with identical deceleration, the greater mass of the loaded truck requires proportionally more force.
Distractor Analysis: Newton's first law describes inertia—objects maintain their state of motion unless acted upon by a net force. Newton's third law states every action has an equal and opposite reaction. Gravitational law describes the attractive force between masses, unrelated to stopping motion.
Takeaway: In braking scenarios, force required is directly proportional to mass when deceleration is constant, a direct application of F = ma.

Why Correct: Linear momentum conservation holds for all collisions in isolated systems regardless of temperature, as per Newton's third law and the principle of momentum conservation in the absence of external forces.
Distractor Analysis: Temperature measures average kinetic energy of molecules, not conserved in collisions as energy transfers occur. Velocity changes for individual balls during collision due to forces. Kinetic energy conserves only in perfectly elastic collisions, not guaranteed by equal temperature.
Takeaway: Momentum conservation applies to both elastic and inelastic collisions, while kinetic energy conservation distinguishes elastic collisions where no energy converts to heat or deformation.

Why Correct: Newton's second law states F = ma, so mass m = F/a, making applied force/acceleration the correct definition.
Distractor Analysis: velocity/acceleration gives time, not mass. applied force/velocity defines damping coefficient in viscous systems. applied force/increased in momentum defines inverse of velocity change.
Takeaway: Inertial mass resists acceleration, while gravitational mass attracts other masses—both are experimentally equivalent.

Core Formula/Logic: In a perfectly elastic collision, kinetic energy is conserved. The ball's speed magnitude just before impact equals its speed magnitude just after impact, allowing it to return to the original height.
Step-by-Step Solution: 1. The ball falls from height H, converting potential energy mgh to kinetic energy 1/2 mv2. 2. At impact, velocity v = sqrt(2gH). 3. In a fully elastic collision, the ground exerts an impulse that reverses the velocity direction without changing magnitude. 4. The ball rebounds upward with velocity v = sqrt(2gH). 5. It converts kinetic energy back to potential energy, reaching height H again.
Common Pitfall: Assuming energy loss during collision leads to option A (less than before). Thinking elasticity adds energy produces option C (more than before). Ignoring the conservation principle yields option D (height independent of elasticity).
Shortcut/Takeaway: Perfectly elastic collisions conserve both momentum and kinetic energy. For vertical drops, rebound height equals drop height when collision is fully elastic.

Core Formula/Logic: For circular motion in a vertical plane with constant angular velocity ω, tension T = mω2r + mg cosθ at any position, where θ is the angle from the vertical downward direction. Maximum tension occurs when cosθ is maximum (cosθ = 1 at lowest point).
Step-by-Step Solution: 1. At lowest position: θ = 0°, cosθ = 1. T = mω2r + mg(1) = mω2r + mg.
2. At highest position: θ = 180°, cosθ = -1. T = mω2r + mg(-1) = mω2r - mg.
3. At horizontal position: θ = 90°, cosθ = 0. T = mω2r + mg(0) = mω2r.
4. Compare: mω2r + mg > mω2r > mω2r - mg, so tension is maximum at lowest point.
Common Pitfall: Confusing constant angular velocity with constant speed—some mistakenly think tension is uniform throughout the motion, leading to option D. Others incorrectly assume maximum tension occurs at highest point due to centripetal force requirements.
Shortcut/Takeaway: For vertical circular motion with constant ω, tension is maximum at bottom (T = mω2r + mg) and minimum at top (T = mω2r - mg). The mg term adds at bottom, subtracts at top.

Why Correct: In free fall without air resistance, all objects accelerate downward at the same rate (g = 9.8 m/s2) regardless of mass or material composition.
Distractor Analysis: Iron ball hitting earlier would only occur with air resistance due to its higher density and smaller cross-sectional area. Wooden ball hitting earlier is physically impossible in vacuum conditions. Equal probability suggests random chance, but free fall is deterministic under gravity.
Takeaway: Galileo's Leaning Tower of Pisa experiment demonstrated that objects of different masses fall at the same rate in vacuum, disproving Aristotle's theory that heavier objects fall faster.

Core Formula/Logic: Pressure = Force ÷ Area. For a constant weight (force), pressure is inversely proportional to the contact area with the ground.
Step-by-Step Solution: 1. The man's weight (force due to gravity) remains constant in all positions. 2. When lying flat, his body has maximum contact area with the ground. 3. Pressure = Weight ÷ Area, so with maximum area, pressure is minimized.
Common Pitfall: Confusing pressure with force leads to thinking running or standing exerts less force, but weight is constant. Forgetting area variation causes selecting standing or sitting, which have smaller contact areas than lying flat.
Shortcut/Takeaway: For constant weight, pressure is lowest when contact area is largest. Lying flat maximizes area, minimizing pressure—a direct application of P = F/A.