Topic 4.1: Gravitational Fields Questions

State Newton's universal law of gravitation.

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.

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

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

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

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.

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.

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.

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.

An asteroid is roughly spherical with mass M ≈ 9.4 × 10²⁰ kg and radius r ≈ 4.7 × 10⁵ m.

Take G = 6.67 × 10⁻¹¹ N m² kg⁻².

Estimate the escape speed from its surface, and comment on what your value suggests about keeping a base built on the asteroid.

A meteoroid, initially at rest very far from a moon, falls freely and strikes the moon's surface.

The moon has mass M = 1.5 × 10²³ kg and radius r = 2.0 × 10⁶ m, and the moon has no atmosphere.

Take G = 6.67 × 10⁻¹¹ N m² kg⁻².

Determine the speed of the meteoroid as it hits the surface.

Astronomers measure the orbital period T and orbital radius r for several moons of a planet.

Sketch the shape of a graph of T² (y-axis) against r³ (x-axis) for these moons, and state what the gradient of the line represents.

Planet Q has a radius twice that of Earth and a mass eight times that of Earth.

The gravitational field strength at the surface of Earth is g.

What is the gravitational field strength at the surface of planet Q?

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.

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.

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.

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.

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 satellite is positioned at a point P that is a distance r from the centre of Earth, where the gravitational field strength is 7.2 N kg⁻¹.

The satellite is then moved to a point Q that is a distance 3r from the centre of Earth.

Assuming Earth behaves as a point mass, what is the difference between the gravitational field strength at P and at Q?