Newton’s laws of gravitation：練習題

Two point masses, M and 4M, are fixed at a distance L apart. A third point mass m is placed at a point on the line joining the centers of M and 4M such that the resultant gravitational force on m is zero. What is the distance of m from the center of mass M?

Two stationary spheres, each of mass \( M \), are separated by a distance \( 2d \). A small test mass \( m \) is placed at the midpoint of the line joining the centers of the two spheres. If the mass \( m \) is displaced by a small distance \( x \) along the perpendicular bisector of the line joining the two spheres, which of the following is the best expression for the magnitude of the initial acceleration of the mass \( m \)? (Assume \( x \ll d \)).

Two point masses, \( m_1 \) and \( m_2 \), are separated by a distance \( d \) and exert a gravitational force \( F \) on each other. If the mass \( m_1 \) is increased to \( 3 m_1 \) and the distance between the centres of the masses is reduced to \( 0.5 d \), what is the new gravitational force in terms of \( F \)?

A satellite of mass \( m \) is initially in a circular orbit of radius \( R \) around a planet of mass \( M \). An engine is fired to move the satellite into a higher circular orbit of radius \( 2R \). Calculate the work done by the engine on the satellite to achieve this transition.

A binary star system consists of two stars with masses \( M \) and \( 3M \). They orbit their common center of mass in circular paths under their mutual gravitational attraction. If the distance between the centers of the stars is \( d \), what is the orbital period \( T \) of the system?

Two uniform spheres of masses \(M\) and \(9M\) have their centres separated by a fixed distance \(d\). At what distance from the centre of the sphere of mass \(M\), along the line joining their centres, will the net gravitational force on a third mass \(m\) be zero?

A point mass \(m\) is located on the axis of a uniform thin ring of mass \(M\) and radius \(R\), at a distance \(x\) from the centre of the ring. Derive the expression for the magnitude of the gravitational attraction exerted by the ring on the point mass in terms of \(G\), \(M\), \(m\), \(R\), and \(x\).

Two identical solid lead spheres of radius \(R\) and uniform density \(\rho\) are placed in contact with each other. Express the magnitude of the mutual gravitational attraction between the two spheres in terms of the gravitational constant \(G\), \(\rho\), and \(R\).

A binary star system consists of two stars of masses \(M_1\) and \(M_2\) rotating about their common center of mass with a constant separation \(L\).

(a) Explain why the stars must have the same angular velocity \(\omega\) in order for the separation \(L\) to remain constant.

(b) Show that the orbital radius \(r_1\) of the star with mass \(M_1\) is given by \(r_1 = \frac{M_2 L}{M_1 + M_2}\).

(c) Using Newton's law of gravitation and the concept of centripetal force, derive an expression for the period \(T\) of the orbital motion in terms of \(G\), \(M_1\), \(M_2\), and \(L\).