Question 1:
a) A bird flies in the east direction with a speed of 5 ms−1. The wind is blowing in the direction of 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 a constant speed all along a straight line at an angle of 30° with the horizontal. The weight of plane is 80, 000 N and its engine gives a thrust of 100, 000 N in the direction of flight. Two additional 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 the free-body diagram exhibiting all forces on the plane. Find out the lift force and the force due to air resistance.
Question 2: A bus is moving downhill at a slope of 5° on a rainy day. At the moment when the speed of the bus is 30 km h−1, the driver spots a deer 30 m ahead. He applies the brakes and comes to a stop. The deer is paralyzed by fear and doesn’t move. Will the bus stop before reaching it or will it hit the deer? Do relevant computations and draw suitable force diagram. Take the coefficient of kinetic friction to be µk = 0.26.
Question 3: Derive an expression relating to impulse and linear momentum. In a safety test, a car of mass 1000 kg is driven into a brick wall. Its bumper behaves like a spring (k = 5 x 106 Nm−1) and is compressed by a distance of 3 cm as the car comes to rest. Find out the initial speed of the car.
Question 4:
a) Can we move a merry-go-round by applying a force all along the radial direction? Describe.
b) A circular disc rotates on a thin air film with a period of 0.3 s. Its moment of inertia about its axis of rotation is around 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 kg m2. Find out the final period of the rotating disc and mass.
Question 5: Obtain an 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 0.93 g. Compute the period of its orbit. Take g = 9.8 ms−2 and the radius of the earth R = 6.37 × 106 m.
Question 6:
a) State against each observation below whether it is true or false. Give reasons for your answer.
• The angular momentum of an artificial satellite rotating about the earth under its gravitation varies with time.
• An alpha particle scattered from an atomic nucleus moves in a plane.
• 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.
Question 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 × 1037 kg m2. Compute its rotational kinetic energy.
b) Assume that 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 doesn’t matter? Describe assuming that the mechanical energy is conserved.
Question 8:
a) The differential cross-section of a collision process in center-of-mass frame of reference is given by dσ/d? = Ar2 sin θ. What is the total cross-section in the laboratory frame of reference?
b) Describe how the alpha particle scattering experiment gives an estimate of the dimensions of an atomic nucleus.
Question 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 4π × 103 rad s−1. It is located 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.