Chapter 13 Flashcards — Gravitation

Definitions

Q1: What is Newton's law of universal gravitation? A: Every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: \( F = G \frac{m_1 m_2}{r^2} \), where G = 6.674 × 10⁻¹¹ N·m²/kg² is the gravitational constant.

Q2: Define gravitational field and write its equation. A: The gravitational field at a point is the gravitational force per unit mass at that point: \( g = \frac{F}{m} = \frac{GM}{r^2} \). It represents how a mass M influences space around it. Field lines point toward the source mass.

Q3: What is gravitational potential energy? How is it defined? A: Gravitational potential energy is the stored energy in a system of masses due to their separation. For two masses, \( U = -G \frac{m_1 m_2}{r} \), where U = 0 at r = ∞. The negative sign indicates bound systems.

Q4: Define gravitational potential and explain its relationship to potential energy. A: Gravitational potential is potential energy per unit mass: \( \phi = \frac{U}{m} = -G \frac{M}{r} \). It is a scalar field around a mass M. Potential energy of a test mass m in field φ is \( U = m\phi \).

Q5: What is escape velocity? Why is it called "escape"? A: Escape velocity is the minimum speed needed for an object to escape a gravitational field (reach r = ∞ with zero velocity). \( v_{escape} = \sqrt{\frac{2GM}{r}} \). It is "escape" because the object no longer returns; its total energy equals zero.

Q6: What is a gravitational field gradient (tidal force)? A: A tidal force is the differential gravitational force on different parts of an extended object. For an object of size Δr, \( \Delta g = \frac{2GM}{r^3} \Delta r \). This gradient stretches objects aligned with the force direction and compresses them perpendicularly.

Q7: State Kepler's three laws of planetary motion. A:

1. First Law: Planets orbit in ellipses with the Sun at one focus.
2. Second Law: A line from the Sun to a planet sweeps equal areas in equal times.
3. Third Law: The square of the orbital period is proportional to the cube of the semi-major axis: \( T^2 = \frac{4\pi^2}{GM} a^3 \).

Q8: What is orbital velocity for a circular orbit? Derive the relationship. A: Orbital velocity is \( v_{orbit} = \sqrt{\frac{GM}{r}} \). Derivation: Gravitational force provides centripetal force: \( G\frac{Mm}{r^2} = \frac{Mv^2}{r} \). Solving for v gives the result. Higher orbits (larger r) have lower velocities.

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Equations and Applications

Q9: When do you use \( U = -G \frac{m_1 m_2}{r} \) vs. \( U = mgh \)? A: Use \( -G \frac{m_1 m_2}{r} \) for systems where distance r is comparable to or larger than the object's size, or when r changes significantly. Use \( mgh \) only near Earth's surface where g ≈ 10 m/s² is approximately constant (valid for Δr << R_Earth).

Q10: What is the total mechanical energy in a circular orbit? A: For a circular orbit, \( E_{total} = K + U = \frac{1}{2}mv^2 - G\frac{Mm}{r} = -G\frac{Mm}{2r} \). The total energy is negative, indicating a bound orbit. Kinetic energy is +, potential is −, and |U| = 2K.

Q11: How are orbital velocity and escape velocity related? A: Escape velocity is \( \sqrt{2} \) times orbital velocity: \( v_{escape} = \sqrt{2} \cdot v_{orbit} \). At orbital velocity, an object remains bound; at escape velocity, its total energy is zero and it just barely escapes to infinity.

Q12: Given orbital period T and radius r, how do you find the central mass M? A: Use Kepler's third law: \( T^2 = \frac{4\pi^2}{GM} r^3 \). Solving for M: \( M = \frac{4\pi^2 r^3}{GT^2} \). This is how astronomers determine masses of stars and galaxies.

Q13: What is the relationship between orbital period and orbital radius? A: From Kepler's third law, \( T \propto r^{3/2} \). Doubling the orbital radius increases the period by a factor of \( 2^{3/2} \approx 2.83 \). Larger orbits take disproportionately longer.

Q14: Explain the "shell theorem" for gravitational forces. A: A uniform spherical shell of mass exerts no gravitational force on a particle inside it. For a particle outside, the shell acts as if all its mass is concentrated at the center. This justifies treating planets as point masses in orbital calculations.

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Conceptual Questions

Q15: Why does Earth orbit the Sun instead of falling into it? A: Earth has orbital velocity (≈ 30 km/s). Gravity continuously pulls it toward the Sun, but Earth's inertia (tangential velocity) prevents collision. These balance to produce a stable elliptical orbit. Without that velocity, Earth would indeed fall in.

Q16: Does gravitational force depend on the motion of the objects? A: No. The force \( F = G \frac{m_1 m_2}{r^2} \) depends only on masses and separation, not velocity. This is a fundamental property of gravity (unlike magnetic force, which depends on motion). All objects at the same r experience the same force.

Q17: What determines whether an orbit is circular or elliptical? A: An orbit is circular when velocity is exactly \( v_{orbit} = \sqrt{\frac{GM}{r}} \) at all points. An elliptical orbit results when velocity is between this and escape velocity. The shape (eccentricity) depends on the velocity direction and magnitude at a given radius.

Q18: Why does the Moon not escape Earth's gravity? A: The Moon's orbital velocity (≈ 1 km/s) is much less than escape velocity from Earth (≈ 11 km/s). With total energy E = −G Mm/(2r) < 0, the Moon remains bound. It would need to be accelerated to escape speed to leave.

Q19: How do tidal forces affect planetary moons and rings? A: Tidal forces create stress across extended objects. If tidal stress exceeds the object's structural strength, it can be torn apart (Roche limit). Saturn's rings formed when tidal forces destroyed a moon. Ocean tides on Earth result from tidal forces by the Moon and Sun.

Q20: What is the difference between gravitational mass and inertial mass? A: Gravitational mass determines the gravitational force an object produces (source of gravity). Inertial mass determines resistance to acceleration (F = ma). Experiments show they are equal to extraordinary precision. This equivalence is central to Einstein's general relativity.

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Worked Examples

Q21: A satellite orbits Earth at radius r = 2R_E (two Earth radii from center). What is its orbital velocity compared to surface gravity? A: At r = 2R_E: \( v_{orbit} = \sqrt{\frac{GM_E}{2R_E}} = \sqrt{\frac{gR_E}{2}} = \sqrt{\frac{10 \times 6.4 \times 10^6}{2}} \approx 5.6 \) km/s (compared to ≈ 11.2 km/s at surface). Orbital velocity decreases with altitude.

Q22: Calculate Earth's escape velocity at the surface (g ≈ 10 m/s², R_E ≈ 6.4 × 10⁶ m). A: \( v_{escape} = \sqrt{2gR_E} = \sqrt{2 \times 10 \times 6.4 \times 10^6} = \sqrt{1.28 \times 10^8} \approx 11.3 \) km/s. Any object launched faster than this from Earth's surface will escape (neglecting air resistance).

Q23: Mars has radius 0.53 R_E and surface gravity 0.38 g_E. What is the ratio of escape velocities? A: \( \frac{v_{escape,Mars}}{v_{escape,Earth}} = \sqrt{\frac{g_{Mars} R_{Mars}}{g_E R_E}} = \sqrt{0.38 \times 0.53} = \sqrt{0.201} \approx 0.45 \). Mars escape velocity is ≈ 5.0 km/s (about 44% of Earth's).

Q24: The Moon orbits Earth in T = 27.3 days at distance r = 3.84 × 10⁸ m. Calculate Earth's mass. A: \( M_E = \frac{4\pi^2 r^3}{GT^2} = \frac{4\pi^2 (3.84 \times 10^8)^3}{6.67 \times 10^{-11} (27.3 \times 86400)^2} \approx 6.0 \times 10^{24} \) kg. This matches the accepted value, validating Kepler's law.

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