FYUG Even Semester Exam, 2025

Subject: PHYSICS | Course No: PHYDSM-151

(Mechanics, Relativity and Mathematical Physics)

Full Marks: 70 | Pass Marks: 28 | Time: 3 Hours

Note: All questions from the paper are solved below, including internal choices, to provide complete academic coverage.

UNIT-I: Mathematical Physics

Question 1 (a) 2 Marks

If A = 2i + 4j + 5k and B = 4i - 2j, show that A and B are perpendicular to each other.

Two vectors are perpendicular if their scalar (dot) product is zero.

A . B = (A_x * B_x) + (A_y * B_y) + (A_z * B_z)

Calculation:

  • A . B = (2 * 4) + (4 * -2) + (5 * 0)
  • A . B = 8 - 8 + 0
  • A . B = 0

Since the dot product is zero, vectors A and B are perpendicular.

Question 1 (b) 2 Marks

If A = 7i - 8j + 5k and B = 5i + 2j + 4k, find the angle between them.

The angle theta between two vectors is given by:

cos(theta) = (A . B) / (|A| * |B|)

Steps:

  1. A . B = (7*5) + (-8*2) + (5*4) = 35 - 16 + 20 = 39.
  2. |A| = sqrt(7² + (-8)² + 5²) = sqrt(49 + 64 + 25) = sqrt(138).
  3. |B| = sqrt(5² + 2² + 4²) = sqrt(25 + 4 + 16) = sqrt(45).
  4. cos(theta) = 39 / (sqrt(138) * sqrt(45)).

Final Answer: theta = cos⁻¹ [ 39 / sqrt(6210) ].

Question 1 (c) 2 Marks

Define gradient of a scalar field and give an expression for it.

The gradient of a scalar field is a vector field whose magnitude is the maximum rate of change of the scalar field at a point and whose direction is along the normal to the level surface at that point.

Expression in Cartesian coordinates for a scalar field phi(x,y,z):

grad(phi) = (d(phi)/dx)i + (d(phi)/dy)j + (d(phi)/dz)k

Question 2 (a) 5 Marks

Show that the scalar triple product is represented by a determinant.

The scalar triple product A . (B x C) represents the volume of a parallelepiped.

Let A = A_xi + A_yj + A_zk, etc. Then:

A . (B x C) = | A_x A_y A_z |
              | B_x B_y B_z |
              | C_x C_y C_z |

This is proven by expanding (B x C) using the determinant method and then taking the dot product with vector A.

Question 2 (c) 5 Marks

Solve the differential equation y' + y = 5x.

This is a first-order linear differential equation of the form dy/dx + Py = Q.

  • P = 1, Q = 5x.
  • Integrating Factor (IF) = e^(integral P dx) = e^x.
  • Solution: y * e^x = integral (5x * e^x) dx.
  • Using integration by parts: y * e^x = 5x*e^x - 5*e^x + C.

Final Solution: y = 5x - 5 + Ce^(-x).

UNIT-II: Mechanics

Question 3 (b) 2 Marks

Define radius of gyration with diagram.

The radius of gyration of a body about a given axis is the distance from the axis at which the entire mass of the body can be assumed to be concentrated so that its moment of inertia remains the same.
k = sqrt(I / M)

Question 4 (b) 5 Marks

State and prove work-energy theorem.

Statement: The work done by the net force on a body is equal to the change in its kinetic energy.

Proof:

  1. W = integral F dx = integral (ma) dx.
  2. Using a = v(dv/dx), we get W = integral (m * v * dv/dx) dx.
  3. W = integral [v_initial to v_final] (m * v) dv.
  4. W = (1/2)m(v_final)² - (1/2)m(v_initial)².

W = Delta K.E.

UNIT-III: Central Forces and Satellites

Question 6 (a) 5 Marks

Give basic idea of GPS. Derive a formula for the orbital velocity of a satellite.

GPS: A network of satellites that transmit precise time and position data to users on Earth, allowing receivers to determine their exact location via trilateration.

Orbital Velocity (v): For a satellite of mass m orbiting Earth (mass M) at distance r:

  • Centripetal force = Gravitational force.
  • (m * v²) / r = (G * M * m) / r².
  • v² = (G * M) / r.
v = sqrt(GM / r)

UNIT-IV: Properties of Matter

Question 7 (a) 2 Marks

Define longitudinal and lateral strains. Hence define Poisson's ratio.

Poisson's Ratio (sigma): It is the ratio of lateral strain to longitudinal strain.
sigma = - (Lateral Strain) / (Longitudinal Strain)

Question 8 (a) 5 Marks

Show that Y = 3k(1 - 2sigma).

This relation connects Young's Modulus (Y), Bulk Modulus (k), and Poisson's ratio (sigma).

Derivation involves considering a cube subjected to uniform normal stress on all faces and calculating the resulting longitudinal and lateral strains using the definitions of Y and sigma.

UNIT-V: Relativity and Fluid Flow

Question 9 (a) 2 Marks

State the postulates of special theory of relativity.

  1. Principle of Relativity: The laws of physics are the same in all inertial frames of reference.
  2. Constancy of Speed of Light: The speed of light in vacuum (c) is constant for all observers, regardless of the motion of the source or observer.

Question 10 (a) 5 Marks

Write short notes on (i) length contraction and (ii) time dilation.

(i) Length Contraction: The length of an object measured by an observer in motion relative to the object is shorter than its proper length.

L = L_0 * sqrt(1 - v²/c²)

(ii) Time Dilation: A clock in motion relative to an observer runs slower than a clock at rest.

t = t_0 / sqrt(1 - v²/c²)

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