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Phys 324: Classical Mechanics Question Paper
Phys 324: Classical Mechanics
Course:Bachelor Of Science
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATIONS FOR THE AWARD OF
BACHELOR OF SCIENCE (GENERAL) / BACHELOR OF EDUCATION
PHYS 324: CLASSICAL MECHANICS
STREAMS: BED/ BSC (GEN) Y3S2 TIME: 2 HOURS
DAY/DATE: FRIDAY 26/4/2013 11.30 AM – 1.30 PM
INSTRUCTIONS:
Attempt Question ONE and any other TWO Questions.
Question ONE carries 30 Marks and the rest 20 Marks each
Start each Question on a fresh page.
QUESTION ONE
(a) (i) What are constraints? [1 Mark]
(ii) Using relevant examples, distinguish between the different types of constraints.
[4 Marks]
(iii) Mention difficulties introduced by constraints when solving mechanical problems [2 Marks]
(b) Define generalized coordinates and obtain expressions for: - [2 Marks]
(i) Generalised velocities [2 Marks]
(ii) Generalised displacement [2 Marks]
(iii) Generalised force. [3 Marks]
(c) (i) State the Hamilton’s variation principle and use the same to show that the shortest
distance between two points in a plane is a straight line. [6 Marks]
(ii) Set up a Langrarian for a simple pendulum and obtain the equation to describe its
motion. [5 Marks]
(d) (i) What are the advantages of Langranges approach over Newtonian approach in
mechanics? [2 Marks]
(ii) State the principle of least action. [1 Mark]
QUESTION TWO:
(a) Justify the statement that if a coordinate is cyclic in Langrarian, it will also be cyclic in
Hamiltonian. [6 Marks]
(b) Find the equation of motion of one dimensional harmonic oscillator using Hamilton’s
principle. [8 Marks]
(c) Derive the D’Alemberts principal of virtual work. [6 Marks]
QUESTION THREE
(a) Use the variation principle to deduce the Hamilton’s canonical equations. [12 Marks]
(b) For a certain canonical transformation, it is known that:-
Q=v(q^2+p^2 )
F=1/2 (q^2+p^2 ) ?tan?^(-1) q/p+1/2 qQ.
Find P(q,p)and F_((q,o)). [8 Marks]
QUESTION FOUR
(a) Derive the equation of motion for a particle moving under the influence of a central force. [5 Marks]
(b) Express angular momentum L ? of a system of particles as the sum of angular momentum of motion of the centre mass and angular momentum of motion about the centre of mass.
[10 Marks]
(c) Show that the total angular momentum of a system of particles is conserved if the total external torque N^((e))=0 [5 Marks]
QUESTION FIVE.
(a) State the Newton’s laws of motion. [3 Marks]
(b) Obtain the equation of motion for a particle failing vertically under the influence of gravity when frictional forces obtainable from dissipation function ?kv?^2 are present. Integrate the equation to obtain the velocity as a function of time and show that the maximum possible velocity for fall from rest is v=mg/k. [8 marks]
(c) Two particles of masses m1 and m2 are located on a frictionless double inclined plane and connected by an inextensible massless string, passing over a smooth peg. Use the principle of virtual work to show that for equilibrium, we must have
( sin?_1)/( sin???_2 ? )=m_2/m_1 [9 Marks]
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