Sample Paper I

Question 1. If momentum (P), area (A) and time (T) are taken to be fundamental quantities, then energy has the dimensional formula

(a) $[P^1 A^{-1} T^1]$

(b) $[P^2 A^1 T^1]$

(c) $[P^1 A^{-1/2} T^1]$

(d) $[P^1 A^{1/2} T^{-1}]$

Question 2. The average velocity of a particle is equal to its instantaneous velocity. What is the nature of its motion?

Question 3. A Force of $F = (6\hat{i} - 3\hat{j})N$ acts on a mass of 2kg. Find the magnitude of acceleration.

Question 4. The work done by a body against friction always results in

(a) loss of kinetic energy

(b) loss of potential energy

(c) gain of kinetic energy

(d) gain of potential energy.

Question 5. Which of the following points is the likely position of the centre of mass of the system shown in Fig. 1.

(a) A

(b) B

(c) C

(d) D

Question 6. Two molecules of a gas have speeds $9 \times 10^6$ m/s and $1.0 \times 10^6$ m/s, respectively. What is the r.m.s. speed?

Question 7. A particle in S.H.M has displacement x given by $x = 3 \cos (5\pi t + \pi)$ where x is in metres and t in seconds. Where is the particle at t = 0 and t = 1/2 s?

Question 8. When the displacement of a particle in S.H.M. is one-fourth of the amplitude, what fraction of the total energy is the kinetic energy?

Question 9. The displacement of a progressive wave is represented by y = A sin($\omega$t – kx), where x is distance and t is time. Write the dimensional formula of (i) $\omega$ and (ii) k.

Question 10. 100 g of water is supercooled to –10°C. At this point, due to some disturbance mechanised or otherwise some of it suddenly freezes to ice. What will be the temperature of the resultant mixture and how much mass would freeze?

($S_w = 1 \text{cal/g/}^\circ\text{C}$ and $L_{w(Fusion)} = 80 \text{cal/g}$)

OR

One day in the morning to take bath, I filled up 1/3 bucket of hot water from geyser. Remaining 2/3 was to be filled by cold water (at room temperature) to bring the mixture to a comfortable temperature. Suddenly I had to attend to some work which would take, say 5-10 minutes before I can take bath. Now I had two options: (i) fill the remaining bucket completely by cold water and then attend to the work; (ii) first attend to the work and fill the remaining bucket just before taking bath. Which option do you think would have kept water warmer? Explain.

Question 11. Prove the following;

For two angles of projection ‘$\theta$’ and (90-$\theta$) (with horizontal) with same velocity ‘V’

(a) range is the same,

(b) heights are in the ratio: $\tan^2\theta:1$.

Question 12. What is meant by ‘escape velocity’? Obtain an expression for escape velocity of an object projected from the surface of the earth.

Question 13. A fighter plane is flying horizontally at an altitude of 1.5 km with speed 720 km/h. At what angle of sight (w.r.t. horizontal) when the target is seen, should the pilot drop the bomb in order to hit the target?

Question 14. A sphere of radius R rolls without slipping on a horizontal road. A, B, C and D are four points on the vertical line through the point of contact ‘A’ (Fig.2). What are the translational velocities of particles at points A, B, C, D? The velocity of the centre of mass is $V_{cm}$.

Question 15. A thermodynamic system is taken from an original state D to an intermediate state E by the linear process shown in the Fig. 3.

Its volume is then reduced to the original value from E to F by an isobaric process. Calculate the total work done by the gas from D to E to F.

Question 16. A flask contains Argon and Chlorine in the ratio 2:1 by mass. The temperature of the mixture is 37°C. Obtain the ratio of (i) average kinetic energy per molecule and (ii) root mean square speed $V_{rms}$ of the molecules of the two gases. Atomic mass of argon = 39.9u; molecular mass of chlorine = 70.9u.

Question 17. Calculate the root mean square speed of smoke particles of mass $5 \times 10^{-17}$ kg in Brownian motion in air at NTP?

Question 18. A ball with a speed of 9m/s strikes another identical ball at rest such that after collision the direction of each ball makes an angle 30° with the original direction. Find the speed of two balls after collision. Is the kinetic energy conserved in this collision process?

Question 19. Derive a relation for the maximum velocity with which a car can safely negotiate a circular turn of radius r on a road banked at an angle $\theta$, given that the coefficient of friction between the car types and the road is $\mu$.

Question 20. Give reasons for the following:–

(a) A circketer moves his hands backwards while holding a catch.

(b) It is easier to pull a lawn mower than to push it.

(c) A carpet is beaten with a stick to remove the dust from it.

Question 21. A helicopter of mass 1000 kg rises with a vertical acceleration of 15 m s⁻². The crew and the passengers weight 300 kg. Give the magnitude and direction of the

(a) force on the floor by the crew and passengers,

(b) action of the rotor of the helicopter on the surrounding air,

(c) force on the helicopter due to the surrounding air.

Question 22. A woman pushes a trunk on a railway platform which has a rough surface. She applies a force of 100 N over a distance of 10 m. Thereafter, she gets progressively tired and her applied force reduces linearly with distance to 50 N. The total distance through which the trunk has been moved is 20 m. Plot the force applied by the woman and the frictional force, which is 50 N. Calculate the work done by the two forces over 20m.

Question 23. Derive equations of motion for a rigid body rotating with constant angular acceleration ‘$\alpha$’ and initial angular velocity $\omega_o$.

Question 24. Derive an expression for the kinetic energy and potential energy of a statellite orbiting around a planet. A satellite of mass 200kg revolves around a planet of mass $5 \times 10^{30}$ kg in a circular orbit $6.6 \times 10^6$ m radius. Calculate the B.E. of the satellite. $G = 6.6 \times 10^{-11} \text{ Nm}^2\text{/kg}^2$.

Question 25. State and prove Bernoulli’s theroem.

Question 26. Consider a cycle tyre being filled with air by a pump. Let V be the volume of the tyre (fixed) and at each stroke of the pump $\Delta V (<< V)$ of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from $P_1$ to $P_2$?

OR

In a refrigerator one removes heat from a lower temperature and deposits to the surroundings at a higher temperature. In this process, mechanical work has to be done, which is provided by an electric motor. If the motor is of 1kW power, and heat is transferred from -3°C to 27°C, find the heat taken out of the refrigerator per second assuming its efficiency is 50% of a perfect engine.

Question 27. Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, the frequencies are in the ratio 1:2:3:4.

Question 28.

(a) Define coefficient of viscosity and write its SI unit.

(b) Define terminal velocity and find an expression for the terminal velocity in case of a sphere falling through a viscous liquid.

OR

The stress-strain graph for a metal wire is shown in Fig. 4. The wire returns to its original state O along the curve EFO when it is gradually unloaded. Point B corresponds to the fracture of the wire.

(i) Upto what point of the curve is Hooke’s law obeyed?

(ii) Which point on the curve corresponds to the elastic limit or yield point of the wire?

(iii) Indicate the elastic and plastic regions of the stress-strain graph.

(iv) Describe what happens when the wire is loaded up to a stress corresponding to the point A on the graph and then unloaded gradually. In particular explain the dotted curve.

(v) What is peculiar about the portion of the stress-strain graph from C to B? Upto what stress can the wire be subjected without causing fracture?

Question 29. It is a common observation that rain clouds can be at about a kilometre altitude above the ground.

(a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also calculate in km/h. ($g = 10\text{m/s}^2$).

(b) A typical rain drop is about 4mm diameter. Estimate its momentum if it hits you.

(c) Estimate the time required to flatten the drop i.e. time between first contact and the last contact.

(d) Estimate how much force such a drop would exert on you.

(e) Estimate to the order of magnitude force on an umbrella. Typical lateral separation between two rain drops is 5 cm.

(Assume that the umbrella cloth is not pierced through !!)

OR

A cricket fielder can throw the cricket ball with a speed $v_o$. If he throws the ball while running with speed u at an angle $\theta$ to the horizontal, find

(i) The effective angle to the horizontal at which the ball is projected in air as seen by a spectator.

(ii) What will be time of flight?

(iii) What is the distance (horizontal range) from the point of projection at which the ball will land?

(iv) Find $\theta$ at which he should throw the ball that would maximise the horizontal range as found in (iii).

(v) How does $\theta$ for maximum range change if $u > v_o, u = v_o, u < v_o$?

Question 30.

(a) Show that in S.H.M., acceleration is directy proportional to its displacement at a given instant

(b) A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period, $T = 2\pi\sqrt{\frac{m}{A\rho g}}$ where m is mass of the body and $\rho$ is density of the liquid

OR

A progressive wave represented by $y = 5 \sin (100\pi t - 0.4\pi x)$ where y and x are in m, t is in s. What is the

(a) amplitude

(b) wave length

(c) frequency

(d) wave velocity

(e) magnitude of particle velocity.