I Semester M. Sc Degree Examination

PHYSICS: Classical Mechanics

Time: 3 Hours Max. Marks: 80

Answer all questions

1. a) Prove that the sum of PE and KE remains constant for a system of particles

moving under the action of a conservative force.

b) Define centre of mass for a system of particles (10+5)

(OR)

2. a) Starting from D’Alembert’s principle, derive an expression for Lagrange’s

equations of motion.

b) What are constraints? Write their classification. (10+5)

3. a) Derive Hamilton’s canonical equations of motion from variational principle.

b) Obtain Hamiltonian for one dimensional simple harmonic oscillator. (10+5)

(OR)

4. a) Define the term ‘canonical transformation’ and hence derive the condition for a

canonical transformation.

b) Define Poisson bracket and express Hamilton’s equation of motion using Poisson’s

equation. (10+5)

5. a) Using Hamilton –Jacobi method solve one dimensional harmonic oscillator

problem.

b) State and prove parallel axis theorem. (10+5)

(OR)

6. a) Set up Euler equations of motion for a rigid body.

b) What do you mean by inertia? Explain its physical significance. (10+5)

7. a) Deduce

2 2 2 2 4 E p c m c o

b) Discuss the simultaneity in relativity. (10+5)

(OR)

8. a) Deduce Newton’s gravitational theory from Einstein’s field equations.

b) What is meant by principle of covariance? Differentiate between covariance and

invariance. (10+5)

9. Answer any FOUR of the following. (5 x 4=20)

a) Check whether force

2 2 ˆ ˆ

F x xy i y x j (2 5 ) (3 4 )

is conservative or not.

b) Obtain the equation of motion for Atwood machine.

c) Give the significance of Hamiltonian function.

d) Using Poisson brackets show that the following transformation is canonical:

2 cos 2 sin t t Q qe p and P qe p

e) Show that the kinetic energy of a rotating rigid body can be written as

1

2

T J

f) Calculate the inertia tensor for the system of four point masses 2g, 3g, 4g and 6g

located at the points (1,0,0), (1,-1,0), (1,1,-1) and (1,-1,1).

g) At what speed does a meter stick move if its length is observed to shrink to 0.5m?

h) Describe the equivalence principle.

PHYSICS: Classical Mechanics

Time: 3 Hours Max. Marks: 80

Answer all questions

1. a) Prove that the sum of PE and KE remains constant for a system of particles

moving under the action of a conservative force.

b) Define centre of mass for a system of particles (10+5)

(OR)

2. a) Starting from D’Alembert’s principle, derive an expression for Lagrange’s

equations of motion.

b) What are constraints? Write their classification. (10+5)

3. a) Derive Hamilton’s canonical equations of motion from variational principle.

b) Obtain Hamiltonian for one dimensional simple harmonic oscillator. (10+5)

(OR)

4. a) Define the term ‘canonical transformation’ and hence derive the condition for a

canonical transformation.

b) Define Poisson bracket and express Hamilton’s equation of motion using Poisson’s

equation. (10+5)

5. a) Using Hamilton –Jacobi method solve one dimensional harmonic oscillator

problem.

b) State and prove parallel axis theorem. (10+5)

(OR)

6. a) Set up Euler equations of motion for a rigid body.

b) What do you mean by inertia? Explain its physical significance. (10+5)

7. a) Deduce

2 2 2 2 4 E p c m c o

b) Discuss the simultaneity in relativity. (10+5)

(OR)

8. a) Deduce Newton’s gravitational theory from Einstein’s field equations.

b) What is meant by principle of covariance? Differentiate between covariance and

invariance. (10+5)

9. Answer any FOUR of the following. (5 x 4=20)

a) Check whether force

2 2 ˆ ˆ

F x xy i y x j (2 5 ) (3 4 )

is conservative or not.

b) Obtain the equation of motion for Atwood machine.

c) Give the significance of Hamiltonian function.

d) Using Poisson brackets show that the following transformation is canonical:

2 cos 2 sin t t Q qe p and P qe p

e) Show that the kinetic energy of a rotating rigid body can be written as

1

2

T J

f) Calculate the inertia tensor for the system of four point masses 2g, 3g, 4g and 6g

located at the points (1,0,0), (1,-1,0), (1,1,-1) and (1,-1,1).

g) At what speed does a meter stick move if its length is observed to shrink to 0.5m?

h) Describe the equivalence principle.

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