D.1 Gravitational fields：練習問題

Newton's law of universal gravitation states that the gravitational force \(F\) between two point masses is inversely proportional to the square of the distance \(r\) between them. If the distance between the two masses is doubled, what is the new gravitational force in terms of \(F\)?

Two point masses, \(M\) and \(4M\), are separated by a distance \(d\). At what distance from the mass \(M\), along the line joining the two centers, is the resultant gravitational field strength equal to zero?

A satellite of mass \(m\) is moved from a stable circular orbit of radius \(R\) to a new stable circular orbit of radius \(2R\) around a planet of mass \(M\). What is the total work done on the satellite during this transition?

A planet has a radius \(R\). At the surface of the planet, the gravitational field strength is \(g\). What is the gravitational field strength at a height above the surface equal to the radius of the planet?

A satellite is in a circular orbit of radius \(r\) around a planet. If the orbital radius is increased to \(4r\), by what factor does the orbital speed of the satellite decrease?

A satellite is in a circular orbit around a planet of mass \(M\) at a distance \(r\) from the center of the planet.

(a) Show that the orbital period \(T\) is proportional to \(r^{3/2}\).
(b) A communications satellite orbits Earth at an altitude of \(3.58 \times 10^7 \text{ m}\). Given the Earth's radius is \(6.37 \times 10^6 \text{ m}\), calculate the gravitational field strength at the satellite's position.
(c) Explain why the satellite is described as being in 'free fall' despite moving in a circle.

Consider a planet of mass \(M\) and radius \(R\). A rocket of mass \(m\) is launched vertically from the surface.

(a) Derive an expression for the minimum work done by the rocket engines to move the rocket from the surface to an altitude equal to the radius of the planet \(R\).
(b) If the rocket is to escape the gravitational pull of the planet entirely from this new altitude (\(2R\) from the center), calculate the required additional velocity.
(c) Sketch a graph showing the variation of gravitational potential \(V\) with distance \(r\) from the center of the planet for \(r \ge R\).