Topic 4.1: Gravitational Fields Questions

State Kepler's second law of planetary motion, and state where in its orbit a planet moves with the greatest speed.

A planet of mass 2.0 × 10³⁰ kg has a small moon in a circular orbit of radius 1.1 × 10¹¹ m.

Calculate the orbital speed of the moon.

Take G = 6.67 × 10⁻¹¹ N m² kg⁻².

State what is meant by the **escape speed** of a planet.

For a planet orbiting the Sun, the orbital period T and orbital radius r are found to obey a relationship of the form Tⁿ ∝ rᵐ, where n and m are whole numbers.

Identify the values of n and m, and hence state the numerical value of the ratio n : m.

A planet has mass M = 6.4 × 10²³ kg and radius r = 3.4 × 10⁶ m.

Take G = 6.67 × 10⁻¹¹ N m² kg⁻².

Calculate the **gravitational potential** at its surface.

State Newton's universal law of gravitation.

Outline Kepler's first law of planetary motion, and describe one way in which the actual orbits of the planets in the Solar System differ from perfect circles.

A satellite travels at constant speed in a circular orbit around a planet.

State the direction of the satellite's acceleration.

Describe, in terms of a **gravitational potential well**, what must happen for a spacecraft resting on a planet's surface to escape the planet completely.

Define gravitational field strength at a point, and state its SI unit.

A planet of radius R produces a gravitational field strength g at its surface.

Two small probes are released from rest above the planet's surface and fall freely (no atmosphere).

Probe X is released from a height R above the surface; probe Y is released from a height 3R above the surface.

How do the probes' accelerations compare at the instant of release and at the moment they reach the surface?

Two small objects, X of mass 1.0 kg and Y of mass 4.0 kg, are released from rest above the same planet.

X is released from a height where the gravitational field strength is 6.0 N kg⁻¹; Y is released from twice the distance from the planet's centre.

A comet follows a highly elliptical orbit around the Sun.

Explain, with reference to Kepler's second law, how and why the comet's speed changes between the point closest to the Sun and the point farthest from the Sun.

For a body in a circular orbit of radius r and period T around a central mass M, show that T² = kr³ where the constant k is equal to 4π²/GM.

At the surface of a planet, a distance 6.0 × 10⁶ m from its centre, the gravitational field strength is 8.0 N kg⁻¹.

Determine the distance from the planet's centre at which the field strength has fallen to 4.0 N kg⁻¹.

Explain how astronomers can determine the mass of the Sun by observing the motion of a planet that orbits it.

A spacecraft is moved from a circular orbit of radius r around a planet to a new circular orbit of radius 4r around the same planet.

The period in the first orbit is 90 minutes.

State Kepler's third law as a proportionality, calculate the period in the new orbit, and state what happens to the spacecraft's orbital speed.

Two small spheres, of mass 3.0 kg and 9.0 kg, are released from rest at the same point high above the Moon's surface, where the gravitational field strength is 1.6 N kg⁻¹.

A planet orbits a star of mass 1.99 × 10³⁰ kg at a distance of 2.3 × 10¹¹ m from the star's centre.

Calculate the gravitational field strength of the star at the planet's orbit.

(G = 6.67 × 10⁻¹¹ N m² kg⁻².)

A new exoplanet survey models the moons orbiting a gas giant. For these moons, the orbital period T and the orbital radius R are found to obey a relationship of the form Tⁿ ∝ Rᵐ that is consistent with Kepler's third law.

Which pair of exponents (n, m) is consistent with Kepler's third law?