Gravitation — Set 6

Correct Answer: B. g = GM/R^2

• **g = GM/R²** = From F = GMm/R² = mg, dividing both sides by m gives g = GM/R². • **Numerical check** — With G = 6.67×10⁻¹¹, M = 6×10²⁴ kg, R = 6.4×10⁶ m: g ≈ 9.8 m/s². • 💡 Wrong-option analysis: g = GMR: wrong dimension (m⁴/s², not m/s²); g = R²/GM: inverted formula giving units of s²/m; g = GM/R: gives m²/s², not m/s².

Correct Answer: A. g' ≈ g(1 - 2h/R)

• **g′ ≈ g(1 − 2h/R)** = From g′ = g·R²/(R+h)², binomial approximation (1+x)⁻² ≈ 1−2x for x = h/R. • **Rate: ≈ −0.003 m/s² per km altitude** — Gravity decreases steadily with increasing height. • 💡 Wrong-option analysis: g(1 + 2h/R): wrong sign — gravity decreases with height; g(1 − h/R): coefficient is 1, not 2; g(1 + h/R): both sign and coefficient are wrong.

Correct Answer: D. Mass is measured in kilograms and does not depend on location

• **Mass in kg, location-independent** = Mass is inertia (amount of matter), a scalar in kilograms, that never changes with location. • **Invariant under any conditions** — An astronaut with mass 70 kg has the same mass on Earth, Moon, or in deep space. • 💡 Wrong-option analysis: Mass increases when g increases: mass and g are completely independent; Measured in newtons and depends on g: newtons is the unit of force/weight; Mass becomes zero in space: in space weight → 0, but mass never changes.

Correct Answer: C. Weight is measured in newtons and depends on g

• **Weight in N, depends on g** = Weight W = mg is a force in newtons; since g varies with location, weight also varies. • **Direction: toward Earth's center** — Weight is a downward vector; mass is a scalar. • 💡 Wrong-option analysis: Measured in kg and location-independent: kg is the unit of mass, not weight; Does not act along any direction: weight always acts toward gravitational center; Always constant: weight changes on different planets and at different heights.

Correct Answer: D. 365.25 days

• **365.25 days** = Earth's orbital period is one year ≈ 365.25 days; the 0.25-day excess causes a leap year every 4 years. • **Earth's orbital speed ≈ 29.8 km/s** — Earth travels about 940 million km per year around the Sun. • 💡 Wrong-option analysis: 10 years: approximately Saturn's orbital period; 30 days: roughly the Moon's period around Earth; 24 hours: Earth's rotation period (one day), not its revolution period.

Correct Answer: A. Equal in magnitude and opposite in direction

• **Equal magnitude, opposite direction** = Newton's third law applies to gravity: if Earth pulls you with force F, you pull Earth with the same F in opposite direction. • **Newton's third law pair** — The forces are always equal regardless of mass difference (Earth vs apple). • 💡 Wrong-option analysis: Zero if one mass is small: F = Gm₁m₂/r² is zero only if mass is literally zero; Unequal if masses differ: Newton's third law guarantees equality regardless of mass ratio; Same direction: equal and opposite means they cannot be in the same direction.

Correct Answer: C. The planet’s mass and radius, not the object’s mass

• **Planet's mass (M) and radius (R)** = v_escape = √(2GM/R); the object's mass m cancels in the energy equation. • **Earth: 11.2 km/s; Moon: 2.4 km/s** — Moon's lower escape speed explains why it has no atmosphere. • 💡 Wrong-option analysis: Only object's mass: it cancels in the formula; Only planet's rotation: provides a small correction but is not the primary factor; Only planet's temperature: temperature does not appear in the escape velocity formula.

Correct Answer: C. 1/sqrt(2) times

• **1/√2 times** = v_e = √(2GM/R); R → 2R gives v_new = √(2GM/2R) = v_original/√2 ≈ 0.707×. • **Decreases with larger radius** — Same mass but twice the radius = weaker surface gravity = lower escape speed. • 💡 Wrong-option analysis: 2 times: would require R to decrease 4-fold; √2 times: would require R to decrease by half; Unchanged: escape velocity depends on R and must change when R doubles.

Correct Answer: C. Radial and directed inward toward the mass

• **Radial, directed inward** = Field lines point toward source mass since gravity is always attractive; they are radially inward for an isolated sphere. • **Line density ∝ field strength** — Lines are denser closer to the mass where g is stronger. • 💡 Wrong-option analysis: Spiral and outward: spiral implies rotation, outward implies repulsion — neither applies; Random direction: spherical symmetry guarantees radial, not random, lines; Circular and outward: circular lines are magnetic; outward implies repulsion.

Correct Answer: A. J/kg

• **J/kg** = Gravitational potential = potential energy per unit mass; energy in joules divided by mass in kg = J/kg. • **Equivalent: m²/s²** — Since 1 J = 1 kg·m²/s², J/kg = m²/s², confirming dimensional analysis. • 💡 Wrong-option analysis: kg m²: unit of moment of inertia; N/kg: equals m/s², unit of gravitational field strength g, not potential V; N m: the joule (energy), but without dividing by kg it is not potential.