What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

State Kepler's three laws of planetary motion.

Law of orbits: planets move in ellipses with the Sun at a focus; law of areas: equal areas are swept in equal times; law of periods: \(T^{2}\propto a^{3}\).

State Newton's universal law of gravitation.

Every body attracts every other with a force proportional to the product of their masses and inversely proportional to the square of the distance, \(F=Gm_1m_2/r^{2}\).

Whose experiment measured G, and what is its value?

Cavendish measured it using a torsion balance; \(G=6.67\times10^{-11}\ \text{N m}^{2}\text{kg}^{-2}\).

How does g vary with height above the surface?

It decreases approximately as \(g(h)=g(1-2h/R_E)\) for small heights.

How does g vary with depth below the surface?

It decreases as \(g(d)=g(1-d/R_E)\), becoming zero at the centre of the Earth.

Write the gravitational potential energy of two masses.

\(U=-\dfrac{Gm_1m_2}{r}\), taken to be zero when the masses are infinitely far apart.

Define escape speed and give its value for Earth.

The minimum projection speed to escape a planet's gravity, \(v_e=\sqrt{2gR_E}\approx11.2\ \text{km s}^{-1}\) for Earth.

Write the orbital speed of a satellite at height h.

\(v=\sqrt{\dfrac{GM_E}{R_E+h}}\); near the surface it equals \(\sqrt{gR_E}\).

Why is the total energy of an orbiting satellite negative?

Because the magnitude of its negative potential energy exceeds its positive kinetic energy, giving \(E=-GmM_E/2(R_E+h)\), the sign of a bound orbit.

What is the gravitational force on a mass inside a uniform spherical shell?

It is zero — the attractions from all parts of the shell cancel exactly.

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In v² = u² + 2as, a is negative for deceleration — keep the sign or the stopping distance comes out wrong.

Chapter 7: Gravitation — Class 11 Physics

Physics · 14 chapters

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Chapter 7: Gravitation

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Kepler's law of areas is a consequence of the conservation of:

Summary

This chapter develops the universal law of gravitation from Kepler's laws of planetary motion: planets move in elliptical orbits with the Sun at a focus (law of orbits); the line joining a planet to the Sun sweeps equal areas in equal times (law of areas, a consequence of conservation of angular momentum); and the square of the orbital period is proportional to the cube of the semi-major axis (law of periods). Newton's universal law states that every body attracts every other with a force \(F=Gm_1m_2/r^{2}\), directed along the line joining them, where \(G=6.67\times10^{-11}\ \text{N m}^{2}\text{kg}^{-2}\) was measured by Cavendish. Acceleration due to gravity \(g=GM_E/R_E^{2}\) decreases with height as \(g(1-2h/R_E)\) and with depth as \(g(1-d/R_E)\). Gravitational potential energy of two masses is \(-Gm_1m_2/r\), taken zero at infinity. The escape speed from Earth is \(v_e=\sqrt{2gR_E}\approx11.2\ \text{km s}^{-1}\). Earth satellites obey Kepler's laws; the orbital speed is \(\sqrt{GM_E/(R_E+h)}\) and the total energy of an orbiting satellite is negative, \(-Gm M_E/2(R_E+h)\), confirming a bound orbit. A uniform spherical shell exerts no gravitational force on a mass inside it.

Kepler's lawsUniversal law of gravitationThe gravitational constantAcceleration due to gravity above and below the surfaceGravitational potential energyEscape speedEarth satellites and energy of an orbiting satellite

Key terms

Kepler's law of areas

The line joining a planet to the Sun sweeps out equal areas in equal time intervals, a consequence of angular momentum conservation.

Universal gravitational constant

The constant \(G=6.67\times10^{-11}\ \text{N m}^{2}\text{kg}^{-2}\) in Newton's law of gravitation, first measured by Cavendish.

Acceleration due to gravity

The acceleration of a freely falling body near a planet's surface, \(g=GM/R^{2}\).

Gravitational potential energy

The energy of a system of masses due to their positions, \(U=-Gm_1m_2/r\), taken zero at infinity.

Escape speed

The minimum speed needed to project a body so that it never returns, \(v_e=\sqrt{2gR}\approx11.2\ \text{km s}^{-1}\) for Earth.

Orbital velocity

The speed of a satellite in a circular orbit, \(v=\sqrt{GM/(R+h)}\).

Important questions

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Kepler's law of areas

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The line joining a planet to the Sun sweeps out equal areas in equal time intervals, a consequence of angular momentum conservation.

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