1. Two planets A and B describe circles of radii r₁ and r₂ round the sun as centre with speed varying inversely as the square root of their radii. Find the angle between the radii of these two planets when their relative angular velocity is zero.

Their angular relative velocity will be zero, if relative linear velocity of B with respect to A is along AB. Or  is parallel to  .

These two vectors are parallel if :

2. If a planet was suddenly stopped in its orbit supposed to be circular, show that it would fall onto the sun in a time times the period of the planet's revolution.

Consider a imaginary planet moving along a strongly extended flat ellipse the extreme points of which are located on the planet's orbit and at the centre of the sun. The semi-major axis of the orbit of such a planet would apparently be half the semi-major axis of the planet's orbit. So the time period of the imaginary planet T' according to Kepler's law will be given by :

3. A thin spherical shell of total mass M and radius R is held fixed. There is a small hole in the shell. A mass m is released from rest  a distance R from the hole along a line that passes through the hole and also through the centre of the shell. This mass subsequently moves under gravitational force of the shell. How long does the mass take to travel from the hole to the point diametrically opposite ?

Let v be the velocity of the particle at point B. Applying conservation of mechanical energy at point A and B, we have

Inside the shell, gravitational field is zero i.e. force on mass m will be zero. Hence the particle will move with constant velocity v. Therefore, the desired time is

4. A satellite is revolving round the earth in a circular orbit of radius r and velocity V_o. A particle is projected from  the satellite in forward direction with relative velocity . Calculate its minimum and maximum distances from earth's centre during subsequent motion of the particle.

       = orbital speed of satellite                     .....  (1)

                       where M = mass of earth

    Absolute velocity of particle would be :

Since v_P lies between orbital velocity and escape velocity, path of the particle would be an ellipse with r being the minimum distance. Let r' be the maximum distance and v'_P its velocity at that moment.

Then from conservation of angular momentum and conservation of mechanical energy, we get :

           mvPr = mv'Pr'              .....  (3)

Solving the above equations, (1), (2) and (3) and (4) we get

Hence the maximum and minimum distances are  and r respectively.

5. Two earth's satellites move in a common plane along circular plane along circular orbits. The orbital radius of one satellite is r = 7000 km while that of the other is 70 km less. What time interval separates the periodic approaches of the satellites to each other over the minimum distance ?

r₁ = 6930 km = 6.93 x 10⁶ m

and  r₂ = 7000 km = 7.0 x 10⁶ m

Now two cases are possible.

Case 1 : When both the satellites are revolving in opposite sense. Then they will be at closest distance after a time, say t₁ when :

Case 2 : When both the satellites are revolving in the same sense. Then they will be at closest distance after a time t₂ when :

Note : If we substitute absolute values of ω₁ and ω₂, answer comes out to be approximately 4.5 days. Hence teh correct answer should be 4.5 days.

6. A body is launched from the earth's surface at an angle α= 30^o to the horizontal at a speed v_o   Neglecting air resistance and earth's rotation. Find the height to which the body will rise. Here M is mass of earth and R the radius of earth.

Let velocity at highest point be v and R + h = r

Applying conservation of angular momentum between P and Q, we have

         mvr  =  mv₀R cos 30^o

Applying conservation of mechanical energy between P and Q. we have

substituting the value of v from equation (1), we get

7. A earth satellite is revolving in a circular orbit of radius 'a' with velocity v₀. A gun is in the satellite and is aimed directly towards the earth. A  bullet is fired from the gun with muzzle velocity . Neglecting resitance offered by cosmic dust and recoil of gun, calculate maximum and minimum distance of bullet from the centre of earth during its subsequent motion.

Orbital speed of satellite is

From conservation of angular momentum at P and Q, we have

                mav₀ = mvr

                    .....     (2)

From conservation of mechanical energy at P and Q, we have :

Substituting values of v and v0 from (1) and (2), we get

8. One of the star of a binary (double) stars  system is rotating in a circular orbit of radius r₁ with time period T. If the mass of this star is m₁. Find the mass m₂ of the other star. Also find distance between the two stars. Take m₁ = 10³⁰ kg, r₁ = 2.455 x 10⁶ km and T = 20l G = 6.67 x 10^-11 N-m²/kg².

For the equation (1 + x)² = 0.59 x³, x ≈ 3

In a double star system both stars revolves about their common centre of mass (COM) with same angular velocity ω but different linear speeds.

For COM

               m₁r₁ = m₂r₂                ..... (1)

               r₁ + r₂ = r                  .....   (2)

The centripetal force for motion of each star is provided by gravitational force. i.e.,

Substituting this in equation (4) we get

      (1  + x)² = 0.59 x³     or  x ≈ 3

Hence mass of second star is

      m₂ = 3m₁ = 3 x 10³⁰ kg

and distance between the two stars is

9. Binary stars of comparable masses m₁ and m₂ rotate under the influence of each other's gravity with a time period T. If they are stopped suddenly in their motions, find their relative velocity when they collide with each other. The radii of the stars are R₁ and R₂ respectively. G is the universal constant of gravitation.

Both the stars rotate about their centre of mass (COM)

For the position of COM

Applying conservation of mechanical energy we have

and v_r = relative velocity between the two stars 

From equation (2) we find that

Substituting the value of r from equation (1) we get

10. The line joining the positions of three identical stars, each of mass M, forms an equilateral triangle of side a. A particle, located at the centroid of the equilateral triangle, is given a velocity v_o in a direction perpendicular to the plane of the triangle. If the particle stops momentarily after travelling a distance 3a, then find v_o.

Total mechanical energy is conserved. Let initial velocity = v_o.

Total mechanical energy

At final position kinetic energy = 0

Body is at a distance of  from each star.

Potential energy = 

Total mechanical energy

From equation (1) and (2) we get,