1. What is Physics? Explain its importance and scope.

Physics is the branch of science that deals with the study of matter, energy, and the interactions between them. It is a fundamental science that seeks to explain the behavior of the universe, ranging from the smallest particles (like atoms) to the largest structures (like galaxies).

Importance and Scope:

• Understanding Nature: Physics helps us understand the fundamental laws that govern the universe.
• Technological Advancements: Many technological innovations, like computers, smartphones, and medical devices, have their foundation in physics principles.
• Applications in Daily Life: Physics plays a crucial role in industries such as construction, transportation, energy, and communications.
• Interdisciplinary Nature: It serves as the foundation for other sciences like chemistry, biology, and engineering.

2. Explain the concept of significant figures and its rules for addition, subtraction, multiplication, and division with examples.

Significant figures: Significant figures are the digits in a number that carry meaningful information about its precision.

Rules:

• Addition/Subtraction: The result should have the same number of decimal places as the least precise number in the operation. (Example: 12.235 + 3.1 = 15.3)
• Multiplication/Division: The result should have the same number of significant figures as the number with the fewest significant figures. (Example: 3.24 × 5.1 = 16.5)

3. What are the SI units of physical quantities? List the base units and derived units with examples.

Base Units:

• Length (meter, m)
• Mass (kilogram, kg)
• Time (second, s)
• Electric Current (ampere, A)
• Temperature (kelvin, K)
• Amount of substance (mole, mol)
• Luminous intensity (candela, cd)

Derived Units:

• Force (newton, N) = kg·m/s²
• Energy (joule, J) = kg·m²/s²
• Pressure (pascal, Pa) = N/m²
• Power (watt, W) = J/s

4. Define displacement, velocity, and acceleration. Derive the equations of motion for an object moving with constant acceleration.

Displacement (s): The shortest distance from the initial to the final position of an object, considering direction.

Velocity (v): The rate of change of displacement.

Acceleration (a): The rate of change of velocity.

Equations of Motion:

• v = u + at
• s = ut + ½ at²
• v² = u² + 2as

5. A car travels with a velocity of 10 m/s for 2 seconds and then accelerates at 4 m/s². What will be its velocity after 5 seconds?

Solution:

• Initial velocity (u) = 10 m/s
• Acceleration (a) = 4 m/s² (after 2 seconds)
• Time (t) = 5 seconds

After 2 seconds, the car travels with a constant velocity of 10 m/s. For the remaining 3 seconds, it accelerates at 4 m/s².

Final velocity: Using v = u + at, v = 10 + 4 × 3 = 10 + 12 = 22 m/s.

6. Define projectile motion. Derive the equations of motion for a projectile.

Projectile Motion: It is the motion of an object that is thrown or projected into the air, influenced only by gravity and its initial velocity.

Equations of motion for a projectile:

• Horizontal motion: \( x = v_0 \cdot t \)
• Vertical motion: \( y = v_0 \cdot \sin(\theta) \cdot t – \frac{1}{2} g \cdot t^2 \)
• Time of flight: \( t = \frac{2v_0 \cdot \sin(\theta)}{g} \)
• Maximum height: \( H = \frac{v_0^2 \cdot \sin^2(\theta)}{2g} \)
• Range: \( R = \frac{v_0^2 \cdot \sin(2\theta)}{g} \)

7. A projectile is thrown at an angle of 45° with a velocity of 20 m/s. Find the range and maximum height.

Solution:

• Initial velocity (v₀) = 20 m/s
• Angle of projection (θ) = 45°
• g = 9.8 m/s²

Range (R): \( R = \frac{v_0^2 \cdot \sin(2\theta)}{g} = \frac{20^2 \cdot \sin(90°)}{9.8} = \frac{400}{9.8} = 40.82 \, \text{m} \)

Maximum Height (H): \( H = \frac{v_0^2 \cdot \sin^2(\theta)}{2g} = \frac{20^2 \cdot \sin^2(45°)}{2 \cdot 9.8} = \frac{400 \cdot 0.5}{19.6} = 10.2 \, \text{m} \)

8. State Newton’s laws of motion. Explain the concept of inertia and momentum.

Newton’s Laws of Motion:

• First Law (Law of Inertia): An object at rest will remain at rest, and an object in motion will remain in motion unless acted upon by an external force.
• Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass: \( F = ma \)
• Third Law: For every action, there is an equal and opposite reaction.

Inertia: It is the resistance of an object to change its state of motion. The greater the mass, the greater the inertia.

Momentum: It is the product of an object’s mass and velocity: \( p = mv \). It is a vector quantity.

9. A car of mass 1000 kg is moving with a velocity of 30 m/s. Calculate the momentum of the car and the force required to stop it in 10 seconds.

Momentum (p): \( p = mv = 1000 \times 30 = 30000 \, \text{kg·m/s} \)

Force (F): Using \( F = \frac{\Delta p}{\Delta t} \), \( F = \frac{30000 – 0}{10} = 3000 \, \text{N} \)

10. Define work and energy. Derive the work-energy theorem.

Work (W): Work is done when a force is applied to an object, and the object moves in the direction of the force.

The formula for work is:
\( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the force applied, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement.

Energy (E): Energy is the capacity to do work.

Work-Energy Theorem: The work done on an object is equal to the change in its kinetic energy:
\( W = \Delta K = \frac{1}{2} m (v^2 – u^2) \), where \( m \) is the mass, \( v \) is the final velocity, and \( u \) is the initial velocity.

11. A person lifts a box of mass 10 kg vertically to a height of 5 meters. Calculate the work done and the potential energy gained by the box.

Given:

• Mass, \( m = 10 \, \text{kg} \)
• Height, \( h = 5 \, \text{m} \)
• Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

Work Done: \[ W = F \cdot h = mgh = 10 \times 9.8 \times 5 = 490 \, \text{J} \]

Potential Energy (PE): \[ PE = mgh = 490 \, \text{J} \]

12. Define the center of mass. Derive the equation for the velocity of the center of mass of a system of particles.

Center of Mass: The center of mass is the point where the total mass of a system can be considered to be concentrated.

Velocity of the Center of Mass: For a system of particles, the velocity of the center of mass is given by: \[ v_{cm} = \frac{\sum m_i v_i}{\sum m_i} \] where \( m_i \) is the mass of the \( i \)-th particle, and \( v_i \) is the velocity of the \( i \)-th particle.

13. State and explain the laws of rotation. Derive the rotational form of Newton’s second law.

First Law (Rotational Inertia): A rotating body will continue rotating at constant angular velocity unless acted upon by an external torque.

Second Law (Rotational Form): The angular acceleration of a body is directly proportional to the applied torque and inversely proportional to its moment of inertia: \[ \tau = I \cdot \alpha \] where \( \tau \) is torque, \( I \) is the moment of inertia, and \( \alpha \) is angular acceleration.

14. State and explain Newton’s law of gravitation. Derive the expression for the gravitational potential energy of an object at height.

Law of Gravitation: Every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: \[ F = \frac{G m_1 m_2}{r^2} \] where \( F \) is the gravitational force, \( G \) is the universal gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between their centers.

Gravitational Potential Energy: The gravitational potential energy of an object at height \( h \) is given by: \[ U = mgh \] where \( m \) is the mass of the object, \( g \) is acceleration due to gravity, and \( h \) is the height.

15. Find the acceleration due to gravity at the surface of the Earth and at a height ‘h’ above the Earth’s surface.

At the Earth’s surface: \[ g = 9.8 \, \text{m/s}^2 \]

At height \( h \) above the Earth’s surface: \[ g_h = g \left( 1 + \frac{h}{R} \right)^2 \] where \( R \) is the radius of the Earth.

16. What is Hooke’s law? Derive the expression for the elastic potential energy stored in a stretched wire.

Hooke’s Law: Hooke’s Law states that the force required to stretch or compress a spring is directly proportional to the displacement, provided the limit of elasticity is not exceeded: \[ F = kx \] where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.

Elastic Potential Energy: The elastic potential energy stored in a stretched wire is given by: \[ U = \frac{1}{2} k x^2 \]

17. Explain the concept of surface tension. Derive the formula for the work done in increasing the surface area of a liquid drop.

Surface Tension: Surface tension is the force per unit length acting along the surface of a liquid, which causes it to contract and resist external forces.

Work Done in Increasing Surface Area: The work done in increasing the surface area of a liquid drop is given by: \[ W = \gamma \Delta A \] where \( \gamma \) is the surface tension, and \( \Delta A \) is the change in surface area.

18. State the first law of thermodynamics. Derive the relation between heat, work, and internal energy.

First Law of Thermodynamics: The change in internal energy of a system is equal to the heat added to the system minus the work done by the system: \[ \Delta U = Q – W \]

19. What is the second law of thermodynamics? Explain the concept of entropy with an example.

Second Law of Thermodynamics: The total entropy of an isolated system always increases over time, and any reversible process increases the entropy of the system.

Entropy: Entropy is a measure of the disorder or randomness of a system. For example, when ice melts into water, the entropy increases.

20. State the ideal gas equation. Derive the kinetic theory of gases and explain the assumptions made in the theory.

Ideal Gas Equation: The ideal gas law is given by: \[ PV = nRT \] where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

Kinetic Theory of Gases: The theory describes the behavior of gases by assuming that gas molecules are in constant random motion. The key assumptions include:

• Gas molecules move in straight lines between collisions.
• Collisions are elastic, meaning no energy is lost.
• The average kinetic energy of gas molecules is proportional to the temperature.