DSE 1 Classical Dynamics

Classical Dynamics

CBCS 2019 : DSE I

Questions

2025 | +3-V-S-CBCS(MS)-Sc(H)-DSE-I-Phy | Full Marks: 80

PART-I | Answer all Questions. | 1x12

1. Degrees of freedom of a particle constrained to move on the surface of sphere is _____.
2. The dimension of generalized force is always dimension of force. True/False
3. Constraint for the system, "rolling of cylinder in an inclined plane without slipping" is holonomic or non-holonomic?
4. If generalised coordinate has the dimension of momentum, then generalised velocity has the dimension of _____.
5. If a co-ordinate corresponding to rotational motion is cyclic, then _____ is a conserved quantity.
6. Work done by virtual displacement is ______.
7. Four dimensional space is called _____ space.
8. The trajectory of a particle in four space is called _____ .
9. If $ds^2=0$, then it is called _____ interval.
10. $p_\mu . p_\mu = $_____.
11. The relative velocity of two photons recede from each other is _____.
12. The expression for Lorentz factor is _____.

PART-II | Answer any eight within two to three sentences | 2x8

1. Under what conditions, Hamiltonian represents the total energy.
2. What is cyclic co-ordinate ?
3. Set up Lagrangian for Atwood's machine.
4. What is holonomic constraint? Give an example of a system for it.
5. Discuss advantages of Hamiltonian mechanism over Lagrangian mechanism.
6. What is four vector ?
7. Find the velocity of a particle having rest mass $m_0$, and kinetic energy equals its rest mass energy.
8. What is mean by space like and time like ?
9. What is Einstein mass-energy relation?
10. State special theory of relativity.

PART-III | Answer any eight of the following (in maximum 75 words.) | 3x8

1. Derive expression for generalized force.
2. Explain generalised co-ordinates.
3. Set up Hamiltonian for linear harmonic oscillator and find equation of motion.
4. Set up Lagrangian for simple pendulum and find equation of motion.
5. Show that shortest distance between two points in plane is straight line.
6. Explain Lorentz fitzgerald contraction.
7. A space ship A moves away from earth with speed 0.9c. Another space ship B is to pass A at a relative speed of 0.6c. Compute the speed of B relative to earth.
8. Discuss twin paradox.
9. What do you understand by a light cone?
10. What is time dilation?

PART-IV | Answer within 500 words each. | 7x4

1. State Hamilton's principle and derive Lagrange's equation of motion from it.
   OR
   Using D Alembert's principle, derive Lagrange's equation of motion for holonomic system.

2. Deduce Hamilton canonical equations and derive the equation of motion for a particle under a central field of force.
   OR
   Discuss calculus of variation and solve brachistochrone problem.

3. Explain relativistic Doppler effect using four vector.
   OR
   Derive expression for four momentum and four force.

4. Derive Lorentz transformation equations considering four dimensional space.
   OR
   Derive expressions for velocity and acceleration four vectors.

2024 | +3-V-S-CBCS(MS)-Sc(H)-DSE-I-Phy

Part-I | Qn 1 | 1x12

• For __ constraints, constraint relations do not explicitly depend on time.
• For _ constraints, constraint relations can be made independent of velocity.
• Lagrange's differential equation is a _ order differential equation.
• The shortest distance between two points on a curved surface is ___.
• For a particle moving with speed of light, the rest mass is ___.
• Newton's second law is _____ under Galilean transformation.
• The rest mass of a particle moving at speed of light is _______.
• The dimension of action integral of Hamilton's principle is ___.
• Fermat's principle of least time is one example of _____ principle.
• According to Minkowski, the fourth coordinate is ______.
• Time dilation leads to the principle of ______.
• For space like interval, the square of the interval is __ than zero.

Part-II | within two to three sentences | 2x8

• Define generalized momentum.
• What is Lagrangian ?
• Write the mathematical form of D' Alembert's principle.
• What is a Brachistochrone ?
• Write the Hamiltonian for a one dimensional harmonic oscillator.
• Write Minkowski space.
• Write Lorentz transformation equation.
• State relativistic Doppler effect.
• At what speed the relativistic mass increases by an amount of 1%.
• Show that the four-dimensional volume element is invariant under Lorentz transformation.

Part-III | in maximum 75 words | 3x8

• Show that the geodesic on a right circular cylinder is a helix.
• Write the expression for a four dimensional position vector.
• State two postulates of special theory of relativity.
• Explain the significance of Lorentz transformations.
• Find the velocity of 1 MeV electron.
• State D'Alembert's principle.
• Write the physical significance of Lagrangian.
• Show that for a conservative system, Hamiltonian is equal to the total energy of the system.
• Using variational principle show that the path of a projectile is a parabola.
• Explain configuration space.

Part-IV | Answer within 500 words each | 7x4

4) Obtain Newton's equation of motion from Lagrange's equation of motion.
OR
Set up the Lagrangian for a compound pendulum which oscillates in a vertical plane about a fixed horizontal axis and find the equation of motion.

5) Obtain the Hamiltonian and hence equation of motion of a charged particle in an electromagnetic field.
OR
Find the equation of motion of a pendulum bob suspended by a spring allowed to swing in a vertical plane.

6) Using Minkowski diagrams, explain space-time diagrams.
OR
Using Lorentz Transformation equations describe Length contraction and time dilation.

7) Define four vectors. Explain it with examples like position four vector and velocity four vector.
OR
Describe Relativistic Kinematics. Then give its application to two body decay of an unstable particle.

2022 | +3-V-S-CBCS(MS)-Sc(H)-DSE-I-Phy-R&B | Full Marks: 80

PART-I | Anwer all questions | 1 × 2

1. Dimension of Hamiltonian is equal to the dimension of ______.
2. The relativistic Doppler effloct holds good for _____ wave.
3. The square of four velocity vector is _____ invariant.
4. According to Minkowski, the fourth coordinate is _____.
5. Fermat's principle of least time is one example of _____ principle.
6. The dimension of action integral of Hamilton's principle is _____ .
7. Shortest distance between two points in a plane is known as ______.
8. $\oint \vec F. \vec{dr} = 0$ does not hold for ____ forces.
9. The constraint of a pendulum with variable lenggh is _____.
10. Lagrangian is a _____ function.
11. The space like interval, the square of the interval is ____ than Zero.
12. Time dialation leads to the principle of _____.

PART-II | Answer any eight within two to three sentences | 2x8

1. What is Minokowski space ?
2. Write Lorentz Transformation equation.
3. State Relativistic Doppler effect.
4. At what speed the relativistic mass increases by an amount of 1%.
5. Show that the four dimensional volume element is invariant under lorentz Transformation.
6. Explain conservation of four Momentum.
7. What is physical significance of Hamiltonian ?
8. What do you mean by Lagrangian ?
9. Explain principle of virtual work.
10. Explain Forces of constraints.
11. Write the advantages of using generalised coordinate.
12. Explain single particle in space.

PART-II | Answer any eight of the following (in maximum 75 words.) | 3×8

1. State D Alembert's principle.
2. Write the physical significance of Lagrangian.
3. Show that for a conservative system Hamiltonian is equal to the total energy of the system.
4. Using variational principle show that the path of a projectile is a parabola.
5. Explain configuration space.
6. Explain Brachistochrone problem with one example.
7. Two photons approach each other. What is there relative velocity ?
8. Write postulates of special theory of Relativity.
9. What are Time like and Light like ?
10. Explain Four velocity and acceleration.

PART-IV | Answer within 500 words each | 7x4

1. Using generalised coordinates, find an expression for virtua, work done in terms of generalised force.
   OR
   Using D Alembert's principle, derive the Lagrange's equation of motion for a particle moving in conservative holonic system.

2. Explain calculus of variation and derive an expression for Euler- Lagranges equation.
   OR
   Discuss Geodesic problem.

3. Using Minkowski diagrams explain space time diagrams.
   OR
   Using Lorentz Transformation equations describe Length contraction and time dialation.

4. Define Four vectors. Explain it with examples like position four vector and velocity four vector.
   OR
   Describe Relativistic kinematics. Then give its application to two body decay of an unstable particle.