Physics · Sample Paper 2

General instructions

Section A · Objective

1. Question 11 mark

   The dimensional formula of work is

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

   The percentage error in measuring the radius of a sphere is 2%. The percentage error in its volume is

   1. A.2%
   2. B.4%
   3. C.6%
   4. D.8%
3. Question 31 mark

   A ball is thrown vertically up with $20{\text{m s}}^{-1}$. The maximum height reached is

   1. A.10 m
   2. B.20 m
   3. C.40 m
   4. D.80 m
4. Question 41 mark

   A body moves in a circle of radius 2 m with a constant speed of $4{\text{m s}}^{-1}$. Its centripetal acceleration is

   1. A.$2{\text{m s}}^{-2}$
   2. B.$4{\text{m s}}^{-2}$
   3. C.$8{\text{m s}}^{-2}$
   4. D.$16{\text{m s}}^{-2}$
5. Question 51 mark

   The coefficient of static friction between a block and a surface is 0.5. The minimum horizontal force needed to move a 4 kg block is

   1. A.10 N
   2. B.20 N
   3. C.40 N
   4. D.8 N
6. Question 61 mark

   A machine does 3000 J of work in 1 minute. Its power is

   1. A.50 W
   2. B.3000 W
   3. C.180 kW
   4. D.300 W
7. Question 71 mark

   The torque of a force $\vec{F}$ acting at position $\vec{r}$ is

   1. A.$\vec{F}\times \vec{r}$
   2. B.$\vec{r}\times \vec{F}$
   3. C.$\vec{r}\cdot \vec{F}$
   4. D.$rF$ always
8. Question 81 mark

   The period of a geostationary satellite is

   1. A.12 h
   2. B.24 h
   3. C.1 h
   4. D.84 min
9. Question 91 mark

   The ratio of stress to strain within the elastic limit is called

   1. A.Poisson's ratio
   2. B.modulus of elasticity
   3. C.yield point
   4. D.compressibility
10. Question 101 mark

    The excess pressure inside a soap bubble of radius $r$ and surface tension $S$ is

    1. A.$\frac{2S}{r}$
    2. B.$\frac{4S}{r}$
    3. C.$\frac{S}{r}$
    4. D.$\frac{S}{2r}$
11. Question 111 mark

    Two rods of the same material and length have radii in the ratio 1 : 2. With the same temperature difference, the ratio of heat flow rates is

    1. A.1 : 2
    2. B.1 : 4
    3. C.2 : 1
    4. D.4 : 1
12. Question 121 mark

    In SHM, the acceleration is maximum at

    1. A.the mean position
    2. B.the extreme positions
    3. C.halfway to the extreme
    4. D.nowhere
13. Question 131 mark

    Assertion (A): A body can have zero velocity and still be accelerating.

    Reason (R): At the highest point of a vertical throw, velocity is zero but acceleration is g.

    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): The pressure of a gas increases when it is heated at constant volume.

    Reason (R): The average kinetic energy of the molecules increases with temperature, so they hit the walls harder and more often.

    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): No heat engine can have an efficiency of 100%.

    Reason (R): Heat flows naturally from a colder body to a hotter body.

    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): Transverse waves cannot travel through gases.

    Reason (R): Gases have no shear (rigidity) modulus.

    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

   Using dimensional analysis, show that the time period of a simple pendulum is $T=k\sqrt{l/g}$.

2. Question 182 marks

   Find the angle between the vectors $\vec{A}=\hat{i}+\hat{j}$ and $\vec{B}=\hat{i}-\hat{j}$.

3. Question 192 marks

   A spring of spring constant $500{\text{N m}}^{-1}$ is compressed by 10 cm. Find the potential energy stored.

4. Question 202 marks

   How much heat is needed to convert 10 g of ice at 0 °C into water at 0 °C? (Latent heat of fusion = $336{\text{J g}}^{-1}$)

5. OR

   Question 202 marks

   Calculate the rms speed of oxygen molecules at 300 K. ($M=0.032{\text{kg mol}}^{-1}$, $R=8.3{\text{J mol}}^{-1}{\text{K}}^{-1}$)

6. Question 212 marks

   A 0.5 kg mass is attached to a spring of constant $200{\text{N m}}^{-1}$. Find the period of its oscillation.

Section C · Short answer

1. Question 223 marks

   A car moving at $20{\text{m s}}^{-1}$ is brought to rest in 40 m by uniform braking. Find (i) the retardation, (ii) the time taken, (iii) the distance covered in the last second.

2. Question 233 marks

   Two blocks of 3 kg and 2 kg are connected by a light string over a frictionless pulley (Atwood machine). Find the acceleration of the system and the tension in the string.

3. Question 243 marks

   State and prove the work–energy theorem for a constant force.

4. Question 253 marks

   A solid sphere rolls without slipping down an incline. Find the ratio of its translational to rotational kinetic energy.

5. Question 263 marks

   Derive an expression for the terminal velocity of a sphere falling through a viscous liquid.

6. Question 273 marks

   Derive an expression for the work done in an isothermal expansion of an ideal gas from volume $V_1$ to $V_2$.

7. OR

   Question 273 marks

   A refrigerator removes 500 J of heat from inside and rejects 700 J to the room. Find the work done on it and its coefficient of performance.

8. Question 283 marks

   A wave is described by $y=0.02\sin (4\pi t-0.5\pi x)$ in SI units. Find the amplitude, frequency, wavelength and speed.

Section D · Case-based

1. Question 294 marks

   Case study — Weighing at different places. A student of mass 50 kg weighs herself at sea level, on top of a mountain and at the bottom of a deep mine. She notices small differences. Take the Earth's radius as 6400 km and $g=10{\text{m s}}^{-2}$ at the surface.

   (i) Does her mass change from place to place? (1 mark)

   (ii) Write the expression for g at a depth d below the surface. (1 mark)

   (iii) Find her weight at a depth of 64 km. (2 marks)

   OR

   (iii) Find her weight at a height of 64 km. (2 marks)

2. Question 304 marks

   Case study — Choosing a crane rope. A crane must lift loads of up to $10^4$ kg. The engineer chooses a steel rope. The elastic limit of steel is $3\times 10^8$ Pa.

   (i) What is meant by elastic limit? (1 mark)

   (ii) Why is steel preferred over copper for such a rope? (1 mark)

   (iii) Find the minimum radius of the rope so that the stress does not exceed the elastic limit. (Take $\pi =3.14$) (2 marks)

   OR

   (iii) Why are crane ropes made of many thin wires twisted together instead of one thick wire? (2 marks)

Section E · Long answer

1. Question 315 marks

   State and prove the theorem of parallel axes. Use it to find the moment of inertia of a thin uniform rod of mass M and length L about an axis through one end, perpendicular to the rod.

2. Question 325 marks

   A body of mass m is whirled in a vertical circle of radius r with a string. Find the minimum speed at the lowest point so that it just completes the circle. Find the tension at the lowest point in that case.

3. OR

   Question 325 marks

   Show that the motion of a simple pendulum for small amplitudes is simple harmonic and derive its time period. Find the length of a seconds pendulum (period 2 s), taking $g={\pi }^2{\text{m s}}^{-2}$.

4. Question 335 marks

   (a) Explain the three modes of heat transfer with an example each. (b) A copper rod 1 m long with cross-section $10^{-4}{\text{m}}^2$ has its ends at 100 °C and 0 °C. Find the rate of heat flow. ($K=400{\text{W m}}^{-1}{\text{K}}^{-1}$)