Sample Paper II

(a) All questions are compulsory.

(b) There are 30 questions in total. Questions 1 to 8 carry one mark each, questions 9 to 18 carry two marks each, questions 19 to 27 carry three marks each and questions 28 to 30 carry five marks each.

(c) There is no overall choice. However, an internal choice has been provided in one question of two marks, one question of three marks and all three questions of five marks each. You have to attempt only one of the given choices in such questions.

(d) Use of calculators is not permitted.

(e) You may use the following physical constants wherever necessary :

$c = 3 \times 10^8 \text{ms}^{-1}$

$h = 6.6 \times 10^{-34} \text{Js}$

$\mu_o = 4\pi \times 10^{–7} \text{ TmA}^{-1}$

Boltzmann constant $k = 1.38 \times 10^{-23} \text{ JK}^{-1}$

Avogadro’s number $N_A = 6.023 \times 10^{23}/\text{mole}$

Question 1. Modulus of rigidity of liquids is

(a) infinity;

(b) zero;

(c) unity;

(d) some finite small non-zero constant value.

Question 2. If all other parameters except the one mentioned in each of the options below be the same for two objects, in which case (s) they would have the same kinetic energy?

(a) Mass of object A is two times that of B.

(b) Volume of object A is half that of B.

(c) Object A if falling freely while object B is moving upward with the same speed at any given point of time.

(d) Object A is moving horizontally with a constant speed while object B is falling freely.

Question 3. If the sun and the planets carried huge amounts of opposite charges,

(a) all three of Kepler’s laws would still be valid.

(b) only the third law will be valid.

(c) the second law will not change.

(d) the first law will still be valid.

Question 4. Which of the following pairs of physical quantities does not have the same dimensional formula?

(a) Work and torque.

(b) Angular momentum and Planck’s constant.

(c) Tension and surface tension.

(d) Impulse and linear momentum.

Question 5. An ideal gas undergoes four different processes from same initial state (Fig.1). Four processes are adiabatic, isothermal, isobaric and isochoric. Out of A, B, C, and D, which one is adiabatic?

(a) (B)

(b) (A)

(c) (C)

(d) (D)

Question 6. Why do two layers of a cloth of equal thickness provide warmer covering than a single layer of cloth of double the thickness?

Question 7. Volume versus temperature graphs for a given mass of an ideal gas are shown in Fig 2 at two different values of constant pressure. What can be inferred about relations between $P_1$ & $P_2$?

(a) $P_1 > P_2$

(b) $P_1 = P_2$

(c) $P_1 < P_2$

(d) data is insufficient.

Question 8. Along a streamline

(a) the velocity of a fluid particle remains constant.

(b) the velocity of all fluid particles crossing a given position is constant.

(c) the velocity of all fluid particles at a given instant is constant.

(d) the speed of a fluid particle remains constant.

Question 9. State Newton’s third law of motion and use it to deduce the principle of conservation of linear momentum.

Question 10. A graph of x v/s t is shown in Fig. 3. Choose correct alternatives from below. (2)

(a) The particle was released from rest at t = 0.

(b) At B, the acceleration a > 0.

(c) At C, the velocity and the acceleration vanish.

(d) Average velocity for the motion between A and D is positive.

(e) The speed at D exceeds that at E.

Question 11. A vehicle travels half the distance L with speed $V_1$ and the other half with speed $V_2$ then its average speed is

(a) $\frac{V_1 + V_2}{2}$

(b) $\frac{V_1 + V_2}{2V_1V_2}$

(c) $\frac{2V_1V_2}{V_1 + V_2}$

(d) $\frac{L(V_1 + V_2)}{V_1V_2}$

Question 12. Which of the diagrams shown in Fig. 4 most closely shows the variation in kinetic energy of the earth as it moves once around the sun in its elliptical orbit?

(a)

(b)

(c)

(d)

Question 13. The vernier scale of a travelling microscope has 50 divisions which coincide with 49 main scale divisions. If each main scale division is 0.5 mm, calculate the minimum inaccuracy in the measurement of distance.

Question 14. A vessel contains two monatomic gases in the ratio 1:1 by mass. The temperature of the mixture is 27°C. If their atomic masses are in the ratio 7:4, what is the (i) average kinetic energy per molecule (ii) r.m.s. speed of the atoms of the gases.

Question 15. A 500kg satellite is in a circular orbit of radius $R_e$ about the earth. How much energy is required to transfer it to a circular orbit of radius $4R_e$? What are the changes in the kinetic and potential energy? ($R_e = 6.37 \times 10^6 \text{ m, } g = 9.8 \text{ ms}^{-2}$)

Question 16. A pipe of 17 cm length, closed at one end, is found to resonate with a 1.5 kHz source. (a) Which harmonic of the pipe resonate with the above source? (b) Will resonance with the same source be observed if the pipe is open at both ends? Justify your answer. (Speed of sound in air = 340 m s⁻¹)

Question 17. Show that the average kinetic energy of a molecule of an ideal gas is directly propotional to the absolute temperature of the gas.

Question 18. Obtain an expression for the acceleration due to gravity at a depth h below the surface of the earth.

Question 19. The position of a particle is given by $r = 6t\hat{i} + 4t^2\hat{j} + 10\hat{k}$ where r is in metres and t in seconds.

(a) Find the velocity and acceleration as a function of time.

(b) Find the magnitude and direction of the velocity at t = 2s.

Question 20. A river is flowing due east with a speed 3m/s. A swimmer can swim in still water at a speed of 4 m/s (Fig. 5).

(a) If swimmer starts swimming due north, what will be his resultant velocity (magnitude and direction)?

(b) If he wants to start from point A on south bank and reach opposite point B on north bank,

(i) which direction should he swim?

(ii) what will be his resultant speed?

(c) From two different cases as mentioned in (a) and (b) above, in which case will he reach opposite bank in shorter time?

Question 21.

(a) A raindrop of mass 1 g falls from rest, from a height of 1 km and hits the ground with a speed of 50 m s⁻¹.

(i) What are the final K.E. of the drop and its initial P.E.?

(ii) How do you account for the difference between the two?

(Take $g = 10ms^{–2}$).

(b) Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V as shown in Fig. 6.

If the collision is elastic, which of the following (Fig. 7) is a possible result after collision?

Question 22. Explain why:

(a) It is easier to pull a hand cart than to push it.

(b) Figure 8 shows (x, t), (y,t) diagrams of a particle moving in 2-dimensions.

If the particle has a mass of 500 g, find the force (direction and magnitude) acting on the particle.

Question 23.

(a) State parallel axis and perpendicular axis theorem.

(b) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be 2MR²/5, where M is the mass of the sphere and R is the radius of the sphere.

Question 24. A 3m long ladder weighing 20 kg leans on a frictionless wall. Its feet rest on the floor 1 m from the wall Find the reaction forces of the wall and the floor.

Question 25. A fully loaded Boeing aircraft has a mass of $3.3 \times 10^5$ kg. Its total wing area is 500 m². It is in level flight with a speed of 960km/h. (a) Estimate the pressure difference between the lower and upper surfaces of the wings. (b) Estimate the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface. (The density of air $\rho = 1.2 \text{ kgm}^{-3}$)

Question 26. Explain briefly the working principle of a refrigerator and obtain an expression for its coefficient of performance.

Question 27. Derive an expression for the apparent frequency of the sound heard by a listener when source of sound and the listener both move in the same direction.

Question 28.

(a) Show that for small amplitudes the motion of a simple pendulum is simple harmonic, hence obtain an expression for its time period.

(b) Consider a pair of identical pendulums, which oscillate independently such that when one pendulum is at its extreme position making an angle of 2° to the right with the vertical, the other pendulum is at its extreme position making an angle of 1° to the left of the vertical. What is the phase difference between the pendulums?

Question 29.

(a) What is capillary rise? Derive an expression for the height to which a liquid rises in a capillary tube of radius r.

(b) Why small drops of a liquid are always spherical in shape.

Question 30.

(a) Derive an expression for the maximum safe speed for a car on a banked track, inclined at angle $\theta$ to the horizontal. $\mu$ is the cofficient of friction between the tracks and the tyres.

(b) A 100 kg gun fires a ball of 1kg from a cliff of height 500 m. It falls on the ground at a distance of 400m from the bottom of the cliff. Find the recoil velocity of the gun. (acceleration due to gravity = 10 m s⁻²)