#### LEVEL – II

Q:1. With what speed should a satellite be projected from earth’s surface so that it starts resolving around earth at a height of 2600 km in circular orbit ? (Radius of earth = 6400 km, g at surface = 9.8 m/sec2)

Ans: 8.99 x 10^{3} km/s

Q:2. A thin spherical shell of radius 3R and mass M and a hollow sphere of mass 3M with R and 2R as internal and external radii are placed concentrically at O. Find the gravitational field & the gravitational potential at Q where OP = 5R/2 and OQ = 4R.

Ans: $ \displaystyle -\frac{G M}{R} $

Q:3. Two small dense stars rotate about their common centre of mass, as a binary system with the period of 1 year for each. One star is of double the mass of the other and the mass of the lighter one is 1/3 the mass of the sun. The distance between the earth and the sun is R. If the distance between two stars is r, then obtain the relation between r and R.

Ans: R = r

Q:4. What is the magnitude of the gravitational force on the particle of mass m due to the rod?

Ans: $ \displaystyle \frac{G M m}{d(L+ d)} $

Q:5. A system consists of a thin ring of radius r and of mass M and a straight wire of linear mass density λ of infinite length placed along the axis of the ring with one of its ends at the centre of the ring. Find the force of interaction between the wire and the ring.

Ans: $ \displaystyle \frac{G M \lambda}{r} $

Q:6. Two massive particles of mass m_{1}and m_{2} are released from rest from a very large distance. Find the speeds of the particles when their distance of separation is r .

Ans: $ \displaystyle v_1 = m_2 \sqrt{\frac{2 G}{(m_1 + m_2)r}} $

$ \displaystyle v_2 = m_1 \sqrt{\frac{2 G}{(m_1 + m_2)r}} $

Q:7. A particle of mass m is kept on the axis of a fixed circular ring of mass M and radius R at a distance x from the centre of the ring. Find the maximum gravitational force between the ring and the particle.

Ans: $ \displaystyle \frac{2 G M m}{3\sqrt{3}R^2} $

Q:8. A double-star, with two stars masses m_{1} and m_{2}, rotates with constant angular speed. If the maximum distance of separation is R, then find the minimum value of angular speed.

Ans: $ \displaystyle \omega = \sqrt{\frac{ G(m_1 + m_2 )}{R^3}} $

Q:9. A projectile is fired vertically upward from the surface of earth with a velocity Kv_{e} where v_{e} is escape velocity and K < 1. Neglecting air resistance, show that the maximum height to which it will rise, measured from the centre of earth, is R/(1- K^{2} ) where R is the radius of earth.

Q:10. A planet of mass m moves along an elliptical orbit around the sun so that its maximum and minimum distances from the sun (mass = M) are equal to r_{1} and r_{2} respectively. Find the angular momentum of this planet relative to the sun.