Old Exam Papers June 2012 (Set 2)

3. An ellipse of axes a and b and a circle of radius b are cut
from the same sheet of thin uniform metal and are
superposed and fixed together with their centres coincident.
The figure is free to move in its own vertical plane about
one end of the major axis; show that the length of the
equivalent simple pendulum is :
5a ab 2b
4a
2 2
4. An uniform sphere of radius a rolls down an inclined rough
plane with inclination equal to with the horizon and the
plane is rough enough to prevent any slipping. Find the
distance travelled by the sphere in time t.
5. An uniform square plate ABCD of mass M and side 2a,
lies on a smooth horizontal plane; it is struck at A by a
particle of mass m moving with velocity V in the direction
of AB, the particle remaining attached to the plate.
Determine the subsequent motion of the system and show
that, its angular velocity is :
m
M 4m
b g .
3V
2a
6. When a body moves under the action of a system
of conservative forces, show that the sum of its
kinetic and potential energies is constant throughout the
motion.
7. A velocity field is given by q xi by tg. j Find the
stream function and the stream line for this field at t = 2.
8. Obtain the equation of continuity in vector form by Euler's
method.
9. Show that the ellipsoid :
2
x
a k t
2 2 2n
F
2
n y z
kt 2
b c
2
2 HG I KJ is a possible form of the boundary surface of a liquid.
10. Steam is rushing from a boiler through a conical pipe the
diameters of the ends of which are D and d. If V and v be
the corresponding velocities of the steam and if the motion
be supposed to be divergent from the vertex of the cone,
prove that v
V
D
d
2 2 2 ev V
j/ 2k
e 2
where k is the pressure divided by density and supposed
constant.
600 2 M.A./M.Sc.-M.T.-05 M.A./M.Sc.-M.T.-05 3 600
1