Work & Energy — Set 4

Correct Answer: A. Zero

• **Zero** = W = F s cosθ; if s = 0 (no displacement), work = 0 regardless of force magnitude or direction. • **W = F × 0 = 0** — pushing against a wall exerts a large force but does zero work since the wall does not move; muscles expend chemical energy but no mechanical work is done on the wall. • 💡 Wrong-option analysis: Equal to force: force and work have different dimensions (N vs J); they can only be numerically equal by accident; Minimum negative: negative work requires displacement opposite to force — with zero displacement there is no work at all; Maximum: zero displacement cannot produce maximum work.

Correct Answer: A. [M L^2 T^-3]

• **[M L² T⁻³]** = power = work/time; work is [M L² T⁻²], dividing by time [T] gives [M L² T⁻³]. • **Watt = J/s = kg·m²/s³** — confirming [M L² T⁻³]; this dimensional formula distinguishes power from energy [M L² T⁻²]. • 💡 Wrong-option analysis: [M L² T⁻²]: this is the dimensional formula for energy (joule), not power; [M L T⁻¹]: this is momentum (kg·m/s); [M L T⁻²]: this is force (newton), not power.

Correct Answer: D. 3000 J

• **3000 J** = work against gravity = weight × height = 600 N × 5 m = 3000 J (weight = mg is already given as 600 N). • **W = 600 × 5 = 3000 J** — this work is stored as gravitational potential energy in the person-Earth system at the top of the stairs. • 💡 Wrong-option analysis: 120 J: 600/5 = 120 — division instead of multiplication; 600 J: this is just the weight — the height (5 m) was not multiplied; 5000 J: unclear arithmetic — perhaps 1000 × 5, using a wrong weight value.

Correct Answer: A. (1/2)kx^2

• **(1/2)kx²** = spring force increases linearly from 0 to kx; average force = (0 + kx)/2 = kx/2; work = average force × displacement = (kx/2) × x = (1/2)kx². • **W = ½kx²** — this equals the elastic potential energy stored in the spring; k is the spring constant (N/m) and x is the extension. • 💡 Wrong-option analysis: (1/2)k²x: this has wrong dimensions — k² × x is [N²/m²]·m = N²/m, not joules; kx²: this is twice the correct answer — the factor ½ arises because force increases linearly from zero; kx: this is the spring force at extension x, not the work done (energy stored).

Correct Answer: D. 4.2 J

• **4.2 J** = one calorie is defined as the heat needed to raise 1 g of water by 1°C; experimentally, 1 cal = 4.186 J ≈ 4.2 J. • **Mechanical equivalent of heat** — Joule's experiments established 1 cal ≈ 4.2 J; food energy is often listed in kilocalories (1 kcal = 4200 J). • 💡 Wrong-option analysis: 42 J: this is 10 calories — ten times too large; 0.42 J: this is 0.1 cal — ten times too small; 4.2 × 10² J = 420 J: this equals 100 calories (0.1 kcal) — 100 times too large.

Correct Answer: A. Can neither be created nor destroyed, only transformed

• **Can neither be created nor destroyed, only transformed** = the law of conservation of energy; in an isolated system the total energy is constant — it merely changes form. • **E_total = const.** — PE converts to KE in free fall, chemical energy to heat in combustion, electrical energy to light in bulbs; the total is always conserved. • 💡 Wrong-option analysis: Always decreases in nature: total energy in an isolated system stays constant; usable energy (exergy) decreases per the second law of thermodynamics, but total energy does not; Can be created from nothing: creating energy from nothing would violate the first law of thermodynamics; Exists only as kinetic energy: energy exists in many forms — potential, thermal, chemical, nuclear, electromagnetic, etc.

Correct Answer: D. -30 J

• **-30 J** = force and displacement are in opposite directions so θ = 180°; W = F s cos 180° = 10 × 3 × (-1) = -30 J. • **cos 180° = -1** — negative work means the force removes kinetic energy from the body; a braking force is a typical example. • 💡 Wrong-option analysis: 30 J: this is |W| — ignoring the direction; work is 30 J in magnitude but negative; 0 J: zero requires θ = 90°; force opposite to displacement means θ = 180°, not 90°; -3 J: this is -F/s = -10/3 — division instead of multiplication.

Correct Answer: B. Zero

• **Zero** = for a conservative force, work done over any closed path is zero because the work depends only on the endpoints (which coincide for a closed path). • **∮ F⃗·ds⃗ = 0** — this is the mathematical condition for conservativeness; equivalently, work depends only on start and end positions. • 💡 Wrong-option analysis: Minimum: the minimum work over a closed path for a conservative force is exactly zero — not a finite minimum; Maximum: work cannot simultaneously be zero (the correct answer) and have a non-zero maximum over a closed path; Infinite: infinite work over a closed path is physically impossible for any standard force.

Correct Answer: B. 98 J

• **98 J** = PE = mgh = 2 × 9.8 × 5 = 98 J. • **PE = 2 × 9.8 × 5 = 98 J** — note g = 9.8 m/s² (not rounded to 10), so 2 × 9.8 = 19.6, then 19.6 × 5 = 98 J. • 💡 Wrong-option analysis: 9.8 J: this is just g — the mass (2 kg) and height (5 m) were not multiplied; 49 J: 2 × 9.8 × 5 / 2 = 49 — incorrectly included an extra factor of ½; 19.6 J: this is m × g = 2 × 9.8 = 19.6 — the height (5 m) was not multiplied.

Correct Answer: C. Four times

• **Four times** = elastic PE = ½kx²; if extension x → 2x, then PE → ½k(2x)² = ½k × 4x² = 4 × ½kx²; it becomes four times. • **PE ∝ x²** — doubling x quadruples the stored energy; this explains why stretching a spring twice as far requires four times the energy. • 💡 Wrong-option analysis: Two times: doubling would require PE ∝ x (linear) — it is actually ∝ x²; One-fourth: one-fourth would occur if x were halved, not doubled; Half: half would occur if x decreased by factor 1/√2.