Work & Energy — Set 7

Correct Answer: B. Negative

• **Negative** = force opposite to displacement means θ = 180°; W = F s cos 180° = -Fs; negative work removes kinetic energy from the body. • **cos 180° = -1** — braking force, friction, and air drag are examples of forces that do negative work by opposing motion. • 💡 Wrong-option analysis: Positive: positive work means force aids displacement, increasing KE — cannot be true if force is opposite to motion; Zero: zero requires θ = 90° (perpendicular), not 180° (opposite); Always maximum: work is minimum (most negative) when force is exactly opposite to displacement — not maximum.

Correct Answer: A. 100 J

• **100 J** = net force along motion = 30 - 10 = 20 N; net work = 20 N × 5 m = 100 J. • **W_net = 20 × 5 = 100 J** — by the work-energy theorem, this equals the change in kinetic energy of the block. • 💡 Wrong-option analysis: 50 J: 10 N × 5 = 50 — uses only the friction force instead of net force; -50 J: this is the work by friction alone (-(10 × 5) = -50 J) — not the net work; 150 J: 30 × 5 = 150 — uses only the applied force without subtracting friction.

Correct Answer: D. erg

• **erg** = the CGS unit of energy; 1 erg = 1 dyne × 1 cm = 10⁻⁷ J (since 1 dyne = 10⁻⁵ N and 1 cm = 10⁻² m). • **1 J = 10⁷ erg** — the erg is a very small unit; it is used in astrophysics and older physics texts but the SI joule is now standard. • 💡 Wrong-option analysis: Watt: the unit of power (SI), not energy — in CGS the power unit would be erg/s; Newton: the SI unit of force — in CGS force is measured in dyne; Joule: the SI unit of energy — in CGS energy is measured in erg.

Correct Answer: A. 30 J

• **30 J** = ΔPE = mg(h₂ - h₁) = 1 × 10 × (5 - 2) = 1 × 10 × 3 = 30 J. • **ΔPE = 1 × 10 × 3 = 30 J** — only the height difference (3 m) matters; the reference level cancels out in ΔPE calculations. • 💡 Wrong-option analysis: 50 J: 1 × 10 × 5 = 50 — uses only the final height, not the height difference (5 - 2 = 3 m); 70 J: 1 × 10 × 7 = 70 — uses (5 + 2 = 7) instead of (5 - 2 = 3); 10 J: 1 × 10 × 1 = 10 — perhaps uses Δh = 1 m by error.

Correct Answer: C. (1/2)k(x2^2 - x1^2)

• **(1/2)k(x₂² - x₁²)** = work done = ΔPE_spring = ½kx₂² - ½kx₁² = ½k(x₂² - x₁²). • **W = ½k(x₂² - x₁²)** — this is the general formula; when x₁ = 0, it reduces to ½kx₂² as expected for stretching from natural length. • 💡 Wrong-option analysis: k(x₂ - x₁): this has wrong dimensions — k(N/m) × (m) = N, not N·m (joules); (1/2)k(x₂ - x₁)²: this expands to ½k(x₂² - 2x₁x₂ + x₁²), which differs from the correct ½k(x₂² - x₁²); kx₂² - kx₁²: this is twice the correct answer — the factor ½ is missing.

Correct Answer: B. P = W/t

• **P = W/t** = power is defined as work done per unit time; the SI unit watt equals one joule per second. • **P = W/t** — rearranged: W = Pt (energy = power × time); this is why kWh = kW × h is a unit of energy, not power. • 💡 Wrong-option analysis: P = W × t: W × t has dimensions J·s — the unit of action (angular momentum), not power; P = W + t: addition of joules (J) and seconds (s) is dimensionally meaningless; P = t/W: t/W has dimensions s/J = 1/W — the inverse of power, not power.

Correct Answer: D. 1 kWh

• **1 kWh** = 30 min = 0.5 h; energy = 2 kW × 0.5 h = 1 kWh. • **E = 2 × 0.5 = 1 kWh** — converting minutes to hours is essential; in joules this is 1 kWh = 3.6 × 10⁶ J. • 💡 Wrong-option analysis: 0.5 kWh: 0.5 kW × 1 h = 0.5 — uses half the power instead of half the time; 2 kWh: 2 kW × 1 h = 2 — uses 1 hour instead of 0.5 hours (30 min); 60 kWh: 2 × 30 = 60 — uses time in minutes (30) instead of hours (0.5).

Correct Answer: A. Thermal energy

• **Thermal energy** = friction is a non-conservative force; its negative work removes mechanical energy from the system and converts it to heat at the sliding surfaces. • **W_friction = -ΔKE** — the kinetic energy lost equals the heat generated; this is why brakes and bearings heat up during operation. • 💡 Wrong-option analysis: Magnetic energy: friction between surfaces does not generate magnetic fields; Gravitational potential energy: friction removes mechanical energy as heat — it does not raise the object to increase PE; Nuclear energy: nuclear energy involves changes at the atomic nucleus level — entirely unrelated to surface friction.

Correct Answer: A. Joule

• **Joule** = the SI unit of work is the joule (J); 1 J = 1 N·m = the work done when 1 newton of force moves an object 1 metre in the direction of the force. • **1 J = 1 kg·m²/s²** — named after James Prescott Joule who established the mechanical equivalent of heat; same unit is used for all forms of energy. • 💡 Wrong-option analysis: Newton: the unit of force (kg·m/s²) — not work; Pascal: the unit of pressure (N/m²) — not work; Watt: the unit of power (J/s) — rate of doing work, not work itself.

Correct Answer: C. Both mass and velocity

• **Both mass and velocity** = KE = ½mv²; kinetic energy depends on mass m (linear) and the square of velocity v² (quadratic), so both are necessary. • **KE = ½mv²** — a heavy slow object and a light fast object can have the same KE; doubling mass doubles KE, but doubling speed quadruples KE. • 💡 Wrong-option analysis: Mass only: KE = ½mv² clearly contains v² — a stationary body (v = 0) has KE = 0 regardless of mass; Height only: height determines potential energy, not kinetic energy; Velocity only: velocity alone is insufficient — KE = ½mv², not ½v².