Gravitational fields：練習問題

A point mass \( m \) is moved from the surface of a planet of mass \( M \) and radius \( R \) to a point at a distance \( 3R \) from the center of the planet. What is the work done by the gravitational field on the mass during this displacement?

A planet has twice the mass of Earth and three times the radius of Earth. If the escape velocity from Earth is \( v_e \), what is the escape velocity from this planet?

A satellite of mass \( m \) is in a stable circular orbit at a height \( h \) above the surface of a planet of radius \( R \) and mass \( M \). Which expression represents the total energy of the satellite?

A satellite is in a circular orbit of radius \(R\) around a planet of mass \(M\). The gravitational potential at this radius is \(V\). What is the kinetic energy of the satellite if its mass is \(m\)?

The gravitational potential \( V \) at the surface of a uniform spherical planet is \( -6.0 \times 10^7 \text{ J kg}^{-1} \). A small mass is projected vertically upwards from the surface with a kinetic energy equal to half the energy required to escape the planet. What is the maximum height reached by the mass, expressed in terms of the planet's radius \( R \)?

A satellite of mass \( m \) is in a circular orbit around a planet of mass \( M \) and radius \( R \). The satellite is at a height \( h \) above the planet's surface.

(a) Derive an expression for the orbital period \( T \) of the satellite in terms of \( G \), \( M \), \( R \), and \( h \).
(b) The satellite is moved to a new geostationary orbit. Explain the three essential characteristics of a geostationary orbit.
(c) A second planet has the same mean density as the first but twice the radius. Show that the escape velocity from the surface of this second planet is twice the escape velocity from the surface of the first planet.