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Classical Mechanics I
MID SEMESTER
26 September, 2006
Duration : 3 hours
Max. Marks: 100
Answer any five questions choosing at least one from (6, 7, 8)
1. (a) Show that
a+b
2
a+b
2
a−b
2
a+b
2
!n
=
an +bn
2
an −bn
2
an −bn
2
an +bn
2
!
for any integer
n > 0.


0 1 0


(b) Find the eigen values and eigen vectors of  1 0 1 
0 1 0
2. Show that the vectors
r̂ = (sinθcosφ, sinθsinφ, cosθ)
θ̂ = (cosθcosφ, cosθsinφ, −sinθ)
φ̂ = (sinφ, −cosφ, 0)
form a set of linearly independent orthonormal vectors.
~ × B)
~ ×C
~ = (A.
~ C)
~ B
~ − (B.
~ C)
~ A.
~
3. (a) Prove (A
~ × B).(
~ C
~ × D)
~ = (A.
~ C)(
~ B.
~ D)
~ − (A.
~ D)(
~ B.
~ C)
~
(b) Prove (A
~ = ~r .
4. (a) Show that ∇r
r
~ 1 ) = − ~r3 .
(b) Show that ∇(
r
r
~ × (∇
~ × ∇)
~ = ∇(
~ ∇.
~ V
~)−∇
~ 2V
~
5. (a) Show that ∇
~ A
~ × B)
~ = (∇
~ × A).
~ B
~ − A.(
~ ∇
~ × B)
~
(b) Show that ∇.(
6. A particle of charge q is conserved in constrained to move in a straight
line between two other equal charges q, fixed at x = ±a. What is the
period of small oscillations? (Mass of the particle is m)
7. (a) Show that the total energy is conserved in one dimensional motion
under a force F (x).
1
(b) A particle of mass m moves under a force F (x) = −cx3 , where c
is a positive constant. Find the potential energy function. If the
particle starts from rest at x = −a, what is the velocity when it
reaches x = 0? Where with subsequent motion does it come to
rest?
8. (a) Show that for an isolated system of N particles mutually interP
secting with each other the total linear momentum N
~i is a
i=1 p
constant of motion.
2
(b) Show that a shift in the origin ~r0 = ~r + ~s, with ddt2~s = 0, does not
~
L
~ , where L
~ is the angular
change the form of the equation dt
=N
~ is the torque.
momentum and N
2