Gravitational fields：練習問題

Which statement best describes gravitational field lines?

A satellite of mass \(m\) orbits a planet of mass \(M\) in a circular orbit of radius \(r\). Which expression gives the speed \(v\) of the satellite?

A small asteroid of mass \(500\) kg is moved from a point where the gravitational potential is \(-1.2 \times 10^8\) J kg\(^{-1}\) to another point where the gravitational potential is \(-2.0 \times 10^8\) J kg\(^{-1}\). What is the work done by the gravitational field?

What is the definition of gravitational field strength at a point?

Which statement correctly defines gravitational potential at a point?

Two point masses, \(m_1 = 4.0 \times 10^3 \, \text{kg}\) and \(m_2 = 6.0 \times 10^3 \, \text{kg}\), are separated by a distance of \(2.5 \, \text{m}\). Calculate the magnitude of the gravitational force between them. (Use \(G = 6.67 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}\))

A test mass is moved from point A on the surface of a planet of mass M and radius R to point B at a distance 3R from the planet's center. Derive an expression for the change in gravitational potential \(\Delta\phi\) between these two points.

A satellite travels from a circular orbit of radius \( R \) to a new stable circular orbit of radius \( 2R \) around a planet of mass \( M \). By considering the expression for gravitational potential \( \phi = -\frac{GM}{r} \), determine the ratio of the gravitational potential at the second orbit to the gravitational potential at the first orbit.

A satellite of mass \(1200 \text{ kg}\) is in a circular orbit around Earth at an altitude of \(600 \text{ km}\) above the Earth's surface.

Given: Mass of Earth \(M_E = 5.97 \times 10^{24} \text{ kg}\)

Radius of Earth \(R_E = 6.37 \times 10^6 \text{ m}\)

Universal gravitational constant \(G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}\)

(a) Calculate the radius of the satellite's orbit from the center of the Earth.

(b) Determine the gravitational force exerted by the Earth on the satellite.

(c) Calculate the gravitational field strength at the satellite's orbital position due to the Earth.

The Earth and the Moon are separated by a center-to-center distance of \( 3.84 \times 10^8 \text{ m} \). The mass of the Earth is \( M_E = 5.97 \times 10^{24} \text{ kg} \) and the mass of the Moon is \( M_M = 7.35 \times 10^{22} \text{ kg} \).
(a) There is a point P on the line joining the centers of the Earth and Moon where the resultant gravitational field strength is zero. Calculate the distance of point P from the center of the Earth.
(b) Sketch a graph showing the variation of the gravitational potential \( \phi \) along the line joining the center of the Earth to the center of the Moon. Label point P on your graph.