Physics · Sample Paper 1

General instructions

Section A · Objective

1. Question 11 mark

   The dimensional formula of pressure is

   1. A.$[MLT^{-2}]$
   2. B.$[ML^{-1}T^{-2}]$
   3. C.$[ML^2T^{-2}]$
   4. D.$[M^{-1}L^3T^{-2}]$
2. Question 21 mark

   The number of significant figures in 0.06900 is

   1. A.2
   2. B.3
   3. C.4
   4. D.5
3. Question 31 mark

   A car starts from rest and accelerates uniformly at $2\ \text{m s}^{-2}$. The distance covered in 5 s is

   1. A.10 m
   2. B.20 m
   3. C.25 m
   4. D.50 m
4. Question 41 mark

   A projectile is fired at $30^\circ$ with the horizontal. For the same speed, the angle that gives the same range is

   1. A.$45^\circ$
   2. B.$60^\circ$
   3. C.$75^\circ$
   4. D.$90^\circ$
5. Question 51 mark

   A force of 10 N acts on a 2 kg body for 3 s. The change in its momentum is

   1. A.$5\ \text{kg m s}^{-1}$
   2. B.$15\ \text{kg m s}^{-1}$
   3. C.$30\ \text{kg m s}^{-1}$
   4. D.$60\ \text{kg m s}^{-1}$
6. Question 61 mark

   If the speed of a body is doubled, its kinetic energy becomes

   1. A.double
   2. B.half
   3. C.four times
   4. D.unchanged
7. Question 71 mark

   The moment of inertia of a thin ring of mass M and radius R about its diameter is

   1. A.$MR^2$
   2. B.$\tfrac12MR^2$
   3. C.$\tfrac25MR^2$
   4. D.$2MR^2$
8. Question 81 mark

   The escape speed from the Earth's surface is about

   1. A.7.9 km/s
   2. B.11.2 km/s
   3. C.3 km/s
   4. D.42 km/s
9. Question 91 mark

   Water flows through a pipe whose cross-section narrows to half. The speed of water in the narrow part becomes

   1. A.half
   2. B.same
   3. C.double
   4. D.four times
10. Question 101 mark

    Water has its maximum density at

    1. A.$0\ ^\circ$C
    2. B.$4\ ^\circ$C
    3. C.$100\ ^\circ$C
    4. D.$-4\ ^\circ$C
11. Question 111 mark

    The rms speed of gas molecules at 27 °C is v. At 927 °C it becomes

    1. A.v
    2. B.2v
    3. C.4v
    4. D.$\sqrt2\,v$
12. Question 121 mark

    A simple pendulum has a period of 2 s. If its length is made four times, the new period is

    1. A.1 s
    2. B.2 s
    3. C.4 s
    4. D.8 s
13. Question 131 mark

    Assertion (A): The work done by the centripetal force on a body in uniform circular motion is zero.

    Reason (R): The centripetal force is always perpendicular to the velocity.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
14. Question 141 mark

    Assertion (A): An astronaut in an orbiting satellite feels weightless.

    Reason (R): There is no gravity at the height of the satellite.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
15. Question 151 mark

    Assertion (A): In an isothermal process the internal energy of an ideal gas does not change.

    Reason (R): The internal energy of an ideal gas depends only on its temperature.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
16. Question 161 mark

    Assertion (A): Sound travels faster in warm air than in cold air.

    Reason (R): The speed of sound in a gas is proportional to the square of its absolute temperature.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.

Section B · Very short answer

1. Question 172 marks

   Check whether the equation $v^2 = u^2 + 2as$ is dimensionally correct.

2. Question 182 marks

   A ball is thrown horizontally at $10\ \text{m s}^{-1}$ from the top of a 20 m high tower. Find the time taken to reach the ground and the horizontal distance covered.

3. Question 192 marks

   State the law of conservation of linear momentum. A 4 kg gun fires a 50 g bullet at $400\ \text{m s}^{-1}$. Find the recoil speed of the gun.

4. Question 202 marks

   A steel wire 2 m long with cross-section $10^{-6}\ \text{m}^2$ is stretched by a 100 N force. Find the extension. ($Y = 2\times10^{11}$ Pa)

5. Question 212 marks

   A particle executes SHM with amplitude 5 cm and period 2 s. Find its maximum speed.

6. OR

   Question 212 marks

   The fundamental frequency of a closed organ pipe is 200 Hz. What are the next two frequencies it can produce?

Section C · Short answer

1. Question 223 marks

   Derive $v^2 = u^2 + 2as$ for uniformly accelerated motion using a velocity–time graph.

2. Question 233 marks

   A 60 kg person stands on a weighing scale in a lift. Find the reading when the lift (i) moves up at constant speed, (ii) accelerates up at $2\ \text{m s}^{-2}$, (iii) accelerates down at $2\ \text{m s}^{-2}$.

3. Question 243 marks

   A 1 kg ball moving at $4\ \text{m s}^{-1}$ collides head-on elastically with an identical ball at rest. Find their velocities after the collision and show that kinetic energy is conserved.

4. Question 253 marks

   State the law of conservation of angular momentum. A dancer spinning at 2 rev/s pulls in her arms so that her moment of inertia becomes half. Find her new speed of rotation. Is kinetic energy conserved?

5. Question 263 marks

   Derive an expression for the orbital speed of a satellite close to the Earth's surface. Calculate it. ($R = 6.4\times10^6$ m)

6. Question 273 marks

   State the law of equipartition of energy. Hence find $C_v$, $C_p$ and $\gamma$ for a monatomic ideal gas.

7. Question 283 marks

   An ideal gas does 200 J of work while 500 J of heat is supplied to it. Find the change in internal energy. If the process were adiabatic with the same work done, what would the change in internal energy be?

8. OR

   Question 283 marks

   State Newton's law of cooling. A body cools from 80 °C to 60 °C in 10 min in a room at 20 °C. Estimate the time it takes to cool from 60 °C to 40 °C.

Section D · Case-based

1. Question 294 marks

   Case study — Aeroplane wings. An aircraft wing is shaped so that air flows faster over its upper surface than under it. By Bernoulli's principle, faster flow means lower pressure, so there is a net upward force called dynamic lift. Take air density $1.2\ \text{kg m}^{-3}$.

   (i) State Bernoulli's principle for streamline flow. (1 mark)

   (ii) If air speeds over and under a wing are $70\ \text{m s}^{-1}$ and $60\ \text{m s}^{-1}$, find the pressure difference. (1 mark)

   (iii) For a wing area of $20\ \text{m}^2$, find the lift force. (2 marks)

   OR

   (iii) Explain why a spinning ball swerves in the air (Magnus effect). (2 marks)

2. Question 304 marks

   Case study — Guitar string. A guitar string 0.5 m long has mass per unit length $0.01\ \text{kg m}^{-1}$ and is under a tension of 100 N. When plucked, it forms standing waves with nodes at both fixed ends.

   (i) Find the speed of transverse waves on the string. (1 mark)

   (ii) Find its fundamental frequency. (1 mark)

   (iii) What happens to the fundamental frequency if the tension is made four times? Explain. (2 marks)

   OR

   (iii) What are beats? Two strings of 100 Hz and 104 Hz are sounded together — how many beats are heard per second? (2 marks)

Section E · Long answer

1. Question 315 marks

   Show that the path of a projectile launched at angle $\theta$ with speed $u$ is a parabola. Derive expressions for its time of flight, maximum height and horizontal range. For what angle is the range maximum?

2. Question 325 marks

   Derive the maximum safe speed of a car on a banked road with friction. Find the optimum speed (no friction needed) on a road of radius 100 m banked at $\tan\theta = 0.4$.

3. Question 335 marks

   Describe the Carnot cycle with a P–V diagram and write the expression for its efficiency. A Carnot engine works between 500 K and 300 K. Find its efficiency and the work done if it absorbs 1000 J of heat.

4. OR

   Question 335 marks

   Derive the expression for the variation of acceleration due to gravity (i) with height and (ii) with depth. At what height above the surface does g become 1% less than at the surface? ($R = 6400$ km)