I. Filling the blanks Two masses m, and m2 exert gravitational forces of equal magnitude on each other. Show that if the total mass M=m1+m2 is fixed, the magnitude of the mutual gravitational force on each of the two masses is a maximum when mI- m2( Fill or= or>) Solution The magnitude of the gravitational force is F_ gm, m, gm,(M-m) dF G(M-2m) The 2. A mass m is inside a uniform spherical shell of mass Mand a mass m is outside the shell as shown in Figure 1. The magnitude of the total gravitational force on m is Mm (s+R)2-d R Solution on nl Fig1 0 F=F F GMm GMm A certain neutron star has a radius of 10.0 km and a mass of 4.00 x 10 kg, about twice the mass of the Sun. The magnitude of the acceleration of an 80.0 kg student foolish enough to be 100 km from the center of the neutron star is 2.67x10 m/s2 The ratio of the magnitude of this acceleration and g is 2. 72x10 If the student is in a circular orbit of radius 100 km about the neutron star, the orbital period is 3.84x10-s Solution (a) The magnitude of the force is F GMm =(x10)y=26×10mII. Filling the Blanks 1. Two masses m1 and m2 exert gravitational forces of equal magnitude on each other. Show that if the total mass M = m1+m2 is fixed, the magnitude of the mutual gravitational force on each of the two masses is a maximum when m1 = m2 ( Fill < or = or > ). Solution: The magnitude of the gravitational force is 2 1 1 2 1 2 ( ) r Gm M m r Gm m Fgrav − = = , We get 2 0 ( 2 ) 0 d d 2 1 1 1 M m r G M m m F = ⇒ = − = ⇒ Then m1 = m2 . 2. A mass m is inside a uniform spherical shell of mass M′ and a mass M is outside the shell as shown in Figure 1. The magnitude of the total gravitational force on m is 2 2 (s R) d Mm F G + − = . Solution: 2 2 2 on ' on total 'on on ( ) 0 s R d GMm r GMm F F F F F F F total total M m M m M m M m + − ∴ = = ∴ = = = + r r r Q r r r 3. A certain neutron star has a radius of 10.0 km and a mass of 4.00 × 1030 kg, about twice the mass of the Sun. The magnitude of the acceleration of an 80.0 kg student foolish enough to be 100 km from the center of the neutron star is 2.67×1010 m/s2 . The ratio of the magnitude of this acceleration and g is 2. 72×109 . If the student is in a circular orbit of radius 100 km about the neutron star, the orbital period is 3.84×10-5 s . Solution: (a) The magnitude of the force is ma r GMm F = = 2 So 10 2 5 2 11 30 2 2.67 10 m/s (1 10 ) 6.67 10 4 10 = × × × × × = = − r GM a M s m M′ R Fig.1 d