TS Intermediate Physics 1st Year Model Paper 2023

SECTION A

Note: (i) Answer ALL Questions

(ii) Each Question carries TWO marks

(iii) All are very short answer type questions. 

1. What is the contribution of S. Chandra Sekhar to Physics?

   Answer: S. Chandrasekhar made significant contributions to astrophysics, particularly in the study of the structure and evolution of stars. His most notable work is the discovery of the Chandrasekhar limit, which states that a white dwarf star can only have a mass up to 1.4 times the mass of the Sun. Beyond this limit, the star will collapse, leading to a supernova. His work on the stability of stars and the dynamics of their evolution earned him the Nobel Prize in Physics in 1983.

2. Can the coefficient of friction be greater than one?

   Answer: Yes, the coefficient of friction can be greater than one. The coefficient of friction (μ) depends on the materials in contact and can be greater than one if the interaction between the materials is strong, such as in rough or sticky surfaces. For example, rubber on dry asphalt can have a coefficient of friction greater than one.

3. When does a real gas behave like an ideal gas?

   Answer: A real gas behaves like an ideal gas at high temperatures and low pressures. At these conditions, the gas molecules are far apart, and intermolecular forces become negligible, so the gas follows the ideal gas law (PV = nRT) closely.

4. State Dalton’s law of partial pressures.

   Answer: Dalton’s law of partial pressures states that the total pressure exerted by a mixture of non-reacting gases is the sum of the partial pressures of the individual gases. Mathematically, P_total = P₁ + P₂ + … + Pn, where P₁, P₂,…, Pn are the partial pressures of the individual gases in the mixture.

5. How can systematic errors be minimized or eliminated?

   Answer: Systematic errors can be minimized or eliminated by improving the experimental setup, using more precise instruments, calibrating instruments properly, ensuring that measurement techniques are consistent, and eliminating or accounting for any bias in the experiment. Repeated measurements and statistical methods can also help reduce systematic errors.

6. Give the expression for the excess pressure in a liquid drop.

   Answer: The excess pressure inside a liquid drop is given by the formula:

   
   $$\Delta P = \frac{4 \sigma}{r}$$

   ΔP=r4σ​

   where ΔP is the excess pressure, σ is the surface tension of the liquid, and r is the radius of the drop.

7. What is the principle behind the carburetor of an automobile?

   Answer: The principle behind the carburetor is based on the Bernoulli’s principle, which states that as the velocity of a fluid increases, the pressure decreases. In a carburetor, air is made to pass through a narrow tube (the venturi), increasing its speed and decreasing its pressure. This low-pressure area draws fuel from the fuel reservoir, mixing it with air, and the resulting fuel-air mixture is then sent to the engine for combustion.

8. Can a substance contract on heating? Give an example.

   Answer: Yes, a substance can contract on heating under certain conditions. For example, water has an anomalous behavior where it contracts between 0°C and 4°C. This is due to the unique structure of water molecules and their hydrogen bonding.

9. What is greenhouse effect? Explain global warming.

   Answer: The greenhouse effect is the process by which certain gases in Earth’s atmosphere (such as carbon dioxide, methane, and water vapor) trap heat from the sun, preventing it from escaping back into space. This process warms the Earth’s surface. Global warming refers to the long-term rise in Earth’s average temperature, primarily caused by the increased concentration of greenhouse gases due to human activities, such as burning fossil fuels, deforestation, and industrial processes.

10. A = i + j. What is the angle between the vector and x-axis?

Answer: The vector A is given as
$A = \hat{i} + \hat{j}$

A=i^+j^​. To find the angle θ between the vector and the x-axis, we use the formula:

$$\cos \theta = \frac{A_x}{|A|}$$

cosθ=∣A∣Ax​​

Here,
$A_x = 1$

(the component along the x-axis), and
$|A| = \sqrt{1^2 + 1^2} = \sqrt{2}$

∣A∣=12+12​=2​.

So,

$$\cos \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

cosθ=2​1​=22​​

Therefore, the angle
$\theta = 45^\circ$

θ=45∘. The angle between the vector A and the x-axis is 45 degrees.

SECTION – B

Note: (i) Answer ANY SIX questions

(ii) Each question carries FOUR marks

(iii) All are of short answer type questions

11. If
    $\vec{A} = \vec{i} – \vec{j}$

    and
    $\vec{B} = \vec{j} – \vec{k}$

    , prove that the angle between
    $\vec{A}$

    and
    $\vec{B}$

    is 90°.

Answer: The angle
$\theta$

between two vectors
$\vec{A}$

and
$\vec{B}$

is given by the formula:

$$\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$$

First, calculate the dot product
$\vec{A} \cdot \vec{B}$

:

$$\vec{A} \cdot \vec{B} = (1)(0) + (-1)(1) + (0)(-1) = -1$$

Next, calculate the magnitudes of
$\vec{A}$

and
$\vec{B}$

:

$$|\vec{A}| = \sqrt{(1)^2 + (-1)^2 + (0)^2} = \sqrt{2}$$

$$|\vec{B}| = \sqrt{(0)^2 + (1)^2 + (-1)^2} = \sqrt{2}$$

Now, substitute into the formula for
$\cos \theta$

:

$$\cos \theta = \frac{-1}{\sqrt{2} \times \sqrt{2}} = \frac{-1}{2}$$

Therefore, the angle
$\theta = 90^\circ$

because the vectors are perpendicular.

12. A ball is dropped vertically from the roof of a tall building and simultaneously another ball is thrown horizontally with some velocity from the same roof. Which ball lands first? Explain your answer.

Answer: Both balls will land at the same time. This is because the horizontal velocity of the ball thrown horizontally does not affect the time it takes for the ball to fall. The time taken for an object to fall under the influence of gravity depends only on its vertical motion, which is the same for both balls. The vertical displacement and acceleration due to gravity are identical for both, so they land at the same time.

13. Explain advantages and disadvantages of friction.

Answer:

• Advantages of Friction:
  • Prevents slipping: Friction between surfaces provides the necessary grip to walk, drive, and hold objects.
  • Enables motion: Friction is necessary in many mechanical systems, such as engines and brakes, to convert energy and control movement.
  • Supports traction: It allows vehicles to move and stop efficiently, and helps in braking systems.
• Disadvantages of Friction:
  • Energy loss: Friction leads to the dissipation of energy in the form of heat, reducing the efficiency of machines and engines.
  • Wear and tear: Continuous friction between moving parts causes wear, leading to damage and the need for maintenance or replacement of parts.
  • Heat generation: Excessive friction can lead to overheating in mechanical systems, causing damage to components.

14. Distinguish between centre of mass and centre of gravity.

Answer:

• Centre of Mass (COM): The centre of mass is the point at which the entire mass of an object or system can be considered to be concentrated. It is a purely physical property that depends on the distribution of mass in the object.
• Centre of Gravity (COG): The centre of gravity is the point at which the total gravitational force acting on an object can be considered to act. It depends on both the mass distribution and the gravitational field acting on the object. For uniform gravitational fields, the centre of gravity coincides with the centre of mass.

15. Define angular acceleration and torque. Establish the relation between angular acceleration and torque.

Answer:

• Angular Acceleration: Angular acceleration (
  $\alpha$

  ) is the rate of change of angular velocity with respect to time. It is measured in radians per second squared (
  $\text{rad/s}^2$

  ).

• Torque: Torque (
  $\tau$

  ) is the rotational equivalent of force. It is the product of the force applied to an object and the perpendicular distance from the axis of rotation to the line of action of the force. It is measured in newton-meters (N·m).

Relation between Angular Acceleration and Torque: The relation between torque and angular acceleration is given by:

$$\tau = I \alpha$$

where
$I$

is the moment of inertia of the object and
$\alpha$

is the angular acceleration. This equation is analogous to Newton’s second law of motion in linear dynamics.

16. What is orbital velocity? Obtain an expression for it.

Answer: Orbital velocity is the velocity required for an object to remain in a stable circular orbit around a planet or other celestial body. It is determined by the gravitational force acting on the object, which provides the necessary centripetal force for circular motion.

The expression for orbital velocity
$v$

is:

$$v = \sqrt{\frac{GM}{r}}$$

where:

• 
  $G$

  is the gravitational constant,

• 
  $M$

  is the mass of the central body (e.g., Earth),

• 
  $r$

  is the radius of the orbit.

17. Define Young’s modulus, Bulk modulus, and Shear modulus.

Answer:

• Young’s Modulus: Young’s modulus (
  $Y$

  ) is a measure of the stiffness of a material. It is defined as the ratio of stress to strain in the linear elastic region of a material under tensile or compressive stress. Its unit is
  $\text{N/m}^2$

  (Pascal).

$$Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$$

• Bulk Modulus: Bulk modulus (
  $K$

  ) is the measure of a material’s resistance to uniform compression. It is defined as the ratio of the change in pressure to the fractional change in volume.

$$K = -V \frac{\Delta P}{\Delta V}$$

• Shear Modulus: Shear modulus (
  $G$

  ) is a measure of the material’s response to shear stress (force applied parallel to the surface). It is defined as the ratio of shear stress to the shear strain.

$$G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$$

18. Pendulum clocks generally go fast in winter and slow in summer. Why?

Answer: Pendulum clocks go fast in winter and slow in summer due to the change in the length of the pendulum caused by temperature variations. In winter, the cooler temperatures cause the material of the pendulum (usually a metal) to contract, shortening its length, which increases its oscillation frequency and causes the clock to run faster. In summer, the warmer temperatures cause the material to expand, lengthening the pendulum, reducing its oscillation frequency, and causing the clock to run slower. The period of a pendulum depends on its length, so temperature-induced changes in length affect the timekeeping.

SECTION – C

Note: (i) Answer ANY TWO questions.

(ii) Each question carries EIGHT marks.

(iii) All are long answer type questions.

19. What are collisions? Explain the possible types of collisions? Develop the theory of one-dimensional elastic collision.

Answer:

• Collisions: Collisions refer to the interaction between two or more bodies where they exert forces on each other for a short time, resulting in a change in their velocities and energy.

• Types of Collisions:

  • Elastic Collision: In an elastic collision, both momentum and kinetic energy are conserved. The objects involved do not experience permanent deformation or heat generation.
  • Inelastic Collision: In an inelastic collision, momentum is conserved, but kinetic energy is not. Some of the energy is converted into other forms, such as heat or sound.
  • Perfectly Inelastic Collision: In a perfectly inelastic collision, the objects stick together after collision. Momentum is conserved, but the maximum amount of kinetic energy is lost.

Theory of One-Dimensional Elastic Collision: For two objects of masses
$m_1$

and
$m_2$

with initial velocities
$u_1$

and
$u_2$

, and final velocities
$v_1$

and
$v_2$

, the law of conservation of momentum and conservation of kinetic energy hold. The equations are:

• Conservation of Momentum:

  
  $$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$

• Conservation of Kinetic Energy:

  
  $$\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$$

These two equations can be solved to find the final velocities
$v_1$

and
$v_2$

after the collision.

20. Show that the motion of a simple pendulum is simple harmonic and hence derive an equation for its time period. What is a seconds pendulum?

Answer:

• Motion of a Simple Pendulum: A simple pendulum consists of a mass
  $m$

  attached to a string of length
  $l$

  and swinging in a gravitational field. For small oscillations (small angles of displacement), the restoring force is proportional to the displacement, which is the characteristic of simple harmonic motion (SHM).

The equation of motion for a simple pendulum is derived from Newton’s second law or from the torque equation. The restoring force is given by:

$$F = -mg \sin \theta$$

For small angles
$\theta$

,
$\sin \theta \approx \theta$

(in radians). The equation becomes:

$$F = -mg \theta$$

The torque (
$\tau$

) is:

$$\tau = -mg l \theta$$

Using the rotational form of Newton’s second law
$\tau = I \alpha$

, where
$I$

is the moment of inertia (
$I = ml^2$

for a point mass) and
$\alpha$

is the angular acceleration (
$\alpha = \frac{d^2 \theta}{dt^2}$

), we get:

$$-mg l \theta = ml^2 \frac{d^2 \theta}{dt^2}$$

Simplifying, we get the equation of SHM:

$$\frac{d^2 \theta}{dt^2} + \frac{g}{l} \theta = 0$$

This is the standard form of SHM with angular frequency
$\omega = \sqrt{\frac{g}{l}}$

.

• Time Period: The time period
  $T$

  of the simple pendulum is the time it takes to complete one full oscillation. For SHM, the time period is given by:

$$T = 2\pi \sqrt{\frac{l}{g}}$$

• Seconds Pendulum: A seconds pendulum is a pendulum whose time period is exactly 2 seconds (i.e., it completes one oscillation in 2 seconds). This corresponds to a pendulum with a length of approximately 1 meter.

21. State second law of thermodynamics.

Answer: The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. In other words, natural processes tend to increase the disorder (entropy) of a system. This law implies that energy spontaneously tends to disperse and become more evenly distributed, making it impossible to construct a perpetual motion machine of the second kind, which would spontaneously convert all heat into work.

Mathematically, the second law can be expressed as:

$$\Delta S_{\text{total}} \geq 0$$

where
$\Delta S_{\text{total}}$

is the change in total entropy of the system and surroundings.

22. How is a heat engine different from a refrigerator?

Answer:

• Heat Engine: A heat engine is a device that converts thermal energy (heat) into mechanical work. It operates by absorbing heat from a high-temperature reservoir, performing work (such as moving a piston), and then releasing some of the heat to a lower-temperature reservoir. A typical example is a steam engine.

• Refrigerator: A refrigerator is a device that transfers heat from a low-temperature reservoir to a high-temperature reservoir, essentially cooling the low-temperature area. It requires work input (typically in the form of electrical energy) to operate the refrigeration cycle, which involves compressing and expanding a refrigerant fluid. The goal of a refrigerator is to maintain a cold temperature in a designated space, like the inside of the refrigerator.