Aptitude Test Problems in Physics Science for Everyone by S Krotov ( PDFDrive.com )

180 Aptitude Test Problems in Physics

1.81. Since the system is closed, the stars will

rotate about their common centre of mass in con-

centric circles. The equations of motion for the

stars have the form

micoT// = F, m 2 412 = F.^ (1)

Here oh and 6) 2 are the angular velocities of rota-

tion of the stars, li and / 2 are the radii of their

orbits, F is the force of interaction between the

stars, equal to Gmi m 2 // 2 , where 1 is the separation

between the stars, and G is the gravitational con-

stant. By the definition of the centre of mass,

m1 1 (^111) = m2 12 , /1 + /2 = /.
Solving Eqs. (1) and (2) together, we obtain

(2)

(01= (02 = V G (mi + m2) =. /... 1 -VG (ml + m2) ,

/ (^3) /
and the required period of revolution of these
stars is
T =2n1 V
1
G (mi +m 2 ) •
1.82. Let v (^1) be the velocity of the station before
the collision, v 2 the velocity of the station and the
meteorite immediately after the collision, m the
mass of the meteorite, and 10m the mass of the
station.
Before the collision, the station moved around
a planet in a circular orbit of radius R. Therefore,
the velocity v 1 of the station can be found from
the equation
=G
10mv 2 10mM
i
R R 2 '
Hence v 1 = j/-GM/R. In accordance with the mo-
mentum conservation law, the velocities u, v„
and v 2 are connected through the following rela-
tion:
mu + 10mv 1 = 11mv2.