Gravitation — Set 5

Correct Answer: D. N m^2 kg^-2

• **N m² kg⁻²** = From G = Fr²/(m₁m₂): [N·m²/kg²], ensuring F = Gm₁m₂/r² is dimensionally consistent. • **Equivalent: m³ kg⁻¹ s⁻²** — Both expressions are correct for G's SI unit. • 💡 Wrong-option analysis: m s⁻²: unit of acceleration g, not G; N kg⁻² m⁻²: wrong power of m — needs m², not m⁻²; kg m² s⁻²: unit of energy (joule), not G.

Correct Answer: A. 9.8 m/s^2

• **g ≈ 9.8 m/s²** = Standard value of free-fall acceleration at Earth's mean surface is 9.80665 m/s². • **Varies 9.76–9.83 m/s²** — Slightly less at equator (9.78) and more at poles (9.83). • 💡 Wrong-option analysis: 3.7 m/s²: surface gravity on Mars; 0 m/s²: g = 0 only at infinite distance or Earth's center; 1.62 m/s²: surface gravity on the Moon.

Correct Answer: D. Gravitational force acting on the body

• **Gravitational force acting on the body** = Weight W = mg is the gravitational pull by Earth (or any planet) on the body, measured in newtons. • **Vector, directed downward** — Mass (kg) is a scalar and location-independent; weight (N) is a vector. • 💡 Wrong-option analysis: Volume occupied: geometric property, not weight; Amount of matter: that is mass in kg; Resistance to change in motion: that is inertia/mass, not gravitational force.

Correct Answer: A. Ellipses with the Sun at one focus

• **Ellipses with Sun at one focus** = Kepler's first law: every planet orbits the Sun in an ellipse with the Sun at one of the two foci. • **Earth's orbital eccentricity ≈ 0.017** — Nearly circular; Mercury's orbit (e = 0.206) is more elongated. • 💡 Wrong-option analysis: Straight lines with constant speed: inertial (force-free) motion, not orbital; Circles with Sun at center: circles are special ellipses but Sun is at focus, not center; Parabolas: parabolic paths are escape trajectories (E = 0), not bound orbits.

Correct Answer: B. Equal areas in equal times

• **Equal areas in equal times** = Kepler's second law: the line joining planet to Sun sweeps equal areas in equal time intervals, reflecting conservation of angular momentum. • **Faster near perihelion** — Earth moves fastest in January (perihelion) and slowest in July (aphelion). • 💡 Wrong-option analysis: Inverse square forces: Newton's law, not Kepler's second law; Constant orbital speed: speed varies along orbit — faster at perihelion; Equal distances in equal times: uniform motion does not apply to elliptical orbits.

Correct Answer: B. It is always attractive

• **Always attractive** = Gravitational force between any two positive masses is always attractive; there is no repulsive gravity in Newtonian mechanics. • **Unlike electromagnetic force** — Electric charges can repel; no negative gravitational mass is observed in nature. • 💡 Wrong-option analysis: Repulsive for large masses: gravity is always attractive regardless of magnitude; Becomes zero if masses are equal: F is zero only if a mass is zero, not when they are equal; Depends on color: gravity depends only on mass and distance.

Correct Answer: A. Gravitational force

• **Gravitational force** = In circular orbit, gravity provides centripetal force: GMm/r² = mv²/r, keeping satellite on its curved path. • **No thrust needed** — Once in orbit, no engine force is required; gravity continuously curves the path. • 💡 Wrong-option analysis: Magnetic force: Earth's magnetic field is far too weak at orbital distances; Air resistance: actually slows satellites and causes orbital decay; Frictional force: no surface contact for an orbiting satellite.

Correct Answer: D. GMm/r^2 = mv^2/r

• **GMm/r² = mv²/r** = Gravity (GMm/r²) equals centripetal force (mv²/r); both sides in N, and simplifying gives v = √(GM/r). • **Fundamental orbit condition** — This balance defines circular orbital motion. • 💡 Wrong-option analysis: GMm = mv²r²: dimensionally wrong — units don't match; GMm/r² = mv²: right side has units of energy (J), not force (N); GMm/r = mv²: equates energies, not forces.

Correct Answer: B. Zero

• **Zero at infinity** = Convention: V = 0 at r = ∞; at any finite distance V = −GM/r < 0. • **Convention, not measurement** — Setting V = 0 at infinity makes all bound states have negative PE. • 💡 Wrong-option analysis: Maximum negative: V is most negative near the mass (r → 0), not at infinity; Equal to g: g is field strength (m/s²), not potential (J/kg); Maximum positive: V is always negative or zero in Newtonian gravity.

Correct Answer: D. Approximately g

• **Approximately g** = In free fall (air resistance ignored), only gravity acts: a = mg/m = g ≈ 9.8 m/s², independent of mass. • **g ≈ 9.8 m/s²** — The body gains 9.8 m/s of speed per second; after 2 s it has fallen about 19.6 m. • 💡 Wrong-option analysis: Acceleration equal to its mass: mass is in kg, acceleration in m/s² — not the same quantity; 0 m/s²: zero acceleration means no net force — not true in free fall; Variable starting at zero: free-fall acceleration is essentially constant near Earth's surface.