1) (a) A bird flies in east direction with the speed of 5 ms^{−1}. Wind is blowing towards north at a speed of 3 ms^{−1}. Find out the relative velocity of the bird with respect to the wind. Draw suitable diagram for solving the problem.
(b) A plane is flying with the constant speed along a straight line at the angle of 30° with horizontal. The weight of plane is 80, 000 N and its engine provides a thrust of 100, 000 N in direction of flight. Two extra forces are exerted on the plane: the lift force perpendicular to the plane’s wings, and the force due to air resistance opposite to the direction of motion. Draw free-body diagram showing all forces on the plane. Find out the lift force and the force due to air resistance.
2) A bus is moving downhill at a slope of 5°on a rainy day. At the moment when speed of the bus is 30 km h^{−1}, driver spots a deer 30 m ahead. He applies the brakes and comes to a stop. The deer is paralyzed by fear and does not move. Will the bus stop before reaching it or will it hit the deer? Do appropriate computations and draw suitable force diagram. Take coefficient of kinetic friction to be μ_{k} = 0.26.
3) Derive the expression relating impulse and linear momentum. In the safety test, a car of mass 1000 kg is driven into a brick wall. Its bumper behaves like a spring (k= 5×106Nm^{−1}) and is compressed by a distance of 3 cm as the car comes to rest. Find out the initial speed of the car.
4) (a) Can we move a merry-go-round by applying the force along the radial direction? Describe.
(b) A circular disc rotates on the thin air film with a period of 0.3 s. Its moment of inertia about its axis of rotation is 0.06 kg m2. A small mass is dropped onto the disc and rotates with it. The moment of inertia of the mass about the axis of rotation is 0.04 kgm2. Find out the final period of the rotating disc and mass.
5) Obtain the expression for the time period of a satellite orbiting the earth. A space shuttle is in a circular orbit at a height of 250 km from the earth’s surface, where the acceleration due to earth’s gravity is 0.93g. Compute the period of its orbit. Take g= 9.8 ms^{−2 }and the radius of the earth R= 6.37 × 10^{6} m.
6. (a) State against each observation below whether it is true or false. Give reasons for your answer.
i) Angular momentum of an artificial satellite rotating about earth under its gravitation varies with time
ii) An alpha particle scattered from the atomic nucleus moves in a plane.
iii) An artificial satellite moves at greater speed when it is nearer the earth.
(b) When is the earth’s orbital motion around the sun fastest and when is it slowest? In which month of the year is the earth closest to the sun? Describe in not more than 50 words.
7) (a) Express the rotational kinetic energy of the earth in terms of its period of rotation. The moment of inertia of the earth about its spin axis is 8.04×10^{37}kg m^{2}. Compute its rotational kinetic energy.
(b) Assume you are designing a cart for carrying goods downhill. To maximize the cart speed, should you design the wheels so that their moments of inertia about their rotation axes are large or small, or it does not matter? Describe assuming that mechanical energy is conserved.
8)(a) The differential cross-section of a collision process in center-of-mass frame of reference is given by dσ/dΩ=Ar^{2} sin2θ. What is the total cross-section in the laboratory frame of reference?
(b) Describe how alpha particle scattering experiment provides an estimate of the dimensions of an atomic nucleus.
9)(a) A bird of mass 1 kg is flying due south at the latitude of 30°N in the northern hemisphere at a speed of 1 ms^{−1}. Find out the Coriolis force acting on it.
(b) A bacteria of mass 2×10^{−24}kg is rotated in a centrifuge at an angular speed of π×103 rad s^{−1}. It is situated at a distance of 5 cm from the axis of rotation. Compute the effective value of g relative to the rotating frame of reference and the net centrifugal force on the bacteria.