Work & Energy — Set 3

Correct Answer: C. 20 J

• **20 J** = W = F s cosθ = 20 × 2 × cos 60° = 20 × 2 × 0.5 = 20 J. • **cos 60° = 0.5** — only the component of the 20 N force along the displacement (20 × 0.5 = 10 N) does work; 10 N × 2 m = 20 J. • 💡 Wrong-option analysis: 40 J: this is F × s = 20 × 2 — ignores the cosθ factor; 10 J: this is F × s × cos 60° using s = 1 m — wrong displacement; 0 J: zero work requires θ = 90°, but here θ = 60°.

Correct Answer: C. Scalar quantity

• **Scalar quantity** = kinetic energy KE = ½mv² has magnitude only and no direction; it is always positive (or zero) regardless of the direction of motion. • **Energy is always scalar** — you can add energies algebraically; unlike velocity or force, there is no direction associated with KE. • 💡 Wrong-option analysis: Dimensionless quantity: KE has dimensions [M L² T⁻²] and units of joule — it is not dimensionless; Tensor quantity: tensors have multiple components (e.g., stress, inertia tensor) — scalars like KE are rank-0 tensors; Vector quantity: KE depends on speed (v²), not velocity, so it has no direction.

Correct Answer: A. Constant

• **Constant** = in uniform circular motion, speed is constant; since KE = ½mv² and both m and v are constant, KE remains constant throughout. • **Speed unchanged** — direction changes continuously but speed does not; since KE depends on v² (not the direction of v⃗), KE stays fixed. • 💡 Wrong-option analysis: Zero: KE is zero only if v = 0; in circular motion the object is always moving; Decreasing: decreasing KE would mean the object is slowing down, which contradicts 'uniform' circular motion; Increasing: increasing KE would mean the object is speeding up — also contradicts constant speed.

Correct Answer: B. p^2/(2m)

• **p²/(2m)** = substitute v = p/m into KE = ½mv²: KE = ½m(p/m)² = ½m × p²/m² = p²/(2m). • **p²/(2m)** — this form is useful when momentum is given directly; for the same momentum, a lighter body has more KE than a heavier one. • 💡 Wrong-option analysis: pm/2: this has dimensions of momentum × mass, not energy; 2p²/m: this is four times the correct answer — the factor ½ in KE gives 1/(2m), not 2/m; p/(2m): this has dimensions of velocity, not energy.

Correct Answer: B. 4 J

• **4 J** = KE = ½mv² = ½ × 0.5 × (4)² = ½ × 0.5 × 16 = 4 J. • **KE = 0.5 × 0.5 × 16 / 2 = 4 J** — note 4² = 16 and ½ × 0.5 = 0.25; 0.25 × 16 = 4 J. • 💡 Wrong-option analysis: 8 J: this uses KE = mv² = 0.5 × 16 = 8 — missing the ½ factor; 2 J: this might use v = 4 instead of v² = 16, giving ½ × 0.5 × 4 = 1 — close but not matching; 6 J: no standard error produces 6 — likely a random guess.

Correct Answer: A. Only the initial and final heights

• **Only the initial and final heights** = gravity is a conservative force; W_gravity = -ΔPE = mg(h₁ - h₂) depends only on the height difference, not the path. • **Path-independence** — whether the body takes a straight vertical path or a zig-zag route, gravity does the same work if the start and end heights are the same. • 💡 Wrong-option analysis: The path taken: only non-conservative forces (like friction) have path-dependent work; gravity is conservative; The color of the body: color is an optical property with no effect on gravitational work; The time taken: gravity's work depends on displacement, not on how quickly the displacement occurs.

Correct Answer: C. Elastic potential energy

• **Elastic potential energy** = the energy stored in a deformed spring or elastic material due to its change in shape, given by PE = ½kx². • **PE_spring = ½kx²** — when the spring is released, this stored energy converts back to kinetic energy; the stiffer the spring (larger k), the more energy stored per unit extension. • 💡 Wrong-option analysis: Nuclear energy: stored in atomic nuclei through binding energy — unrelated to spring deformation; Chemical energy: stored in molecular bonds through chemical reactions — not mechanical deformation; Thermal energy: the random kinetic energy of molecules due to temperature — not the organised mechanical energy of a spring.

Correct Answer: A. Remains constant

• **Remains constant** = conservation of mechanical energy: when only conservative forces act, KE + PE = constant; any decrease in PE becomes KE and vice versa. • **KE + PE = E_mech = const.** — a freely falling body and a pendulum (ignoring air resistance) are classic examples of this conservation. • 💡 Wrong-option analysis: Always decreases: mechanical energy decreases only when non-conservative forces (friction, air resistance) act — not when only conservative forces act; Becomes zero: zero mechanical energy would mean the body is at rest at the reference height — generally not true; Always increases: energy cannot spontaneously increase from within the system.

Correct Answer: A. 80%

• **80%** = efficiency = (useful output / input) × 100% = (400 / 500) × 100% = 80%. • **η = 400/500 × 100 = 80%** — 100 J is lost (to friction, heat, etc.); 80% of the input energy becomes useful work. • 💡 Wrong-option analysis: 90%: 90% would mean 450 J useful output from 500 J input — but useful output is only 400 J; 70%: 70% would mean 350 J useful output — not consistent with the given 400 J; 125%: this would violate energy conservation — output cannot exceed input.

Correct Answer: D. Rate of doing work

• **Rate of doing work** = power P = W/t = rate at which work is done or energy is transferred; SI unit is watt (W = J/s). • **P = W/t = F v** — higher power means the same task is completed faster; a 2000 W motor does twice the work of a 1000 W motor in the same time. • 💡 Wrong-option analysis: Energy per unit mass: that is specific energy (J/kg), not power; Force per unit area: that is pressure (pascal), not power; Work multiplied by time: W × t has dimensions of J·s, called action — not power.