Download End Semester paper

Properties of Matter
END SEMESTER
27 November, 2007
Duration : 3 hours
Max. Marks : 50
1. (a) Nelson stubbed his toe on a cannon ball, which then rolled off
the deck and fell into the ocean. It sank to the bottom of the
ocean, 4000 m below the surface of the water and is still there.
The cannon ball is an iron sphere, with Young’s modulus Y =
1.55 × 1011 and Poison ratio σ = 0.38. If its radius was R = 10
cm when Nelson kicked it and its radius is now R = 10 +dR cm,
estimate dR
R .
(b) Elastic waves from an earthquake are detected at a seismic station. The longitudinal wave (P wave) arrives 5 minutes before
the transverse wave (S wave). Compute the distance of the earthquake epicenter from the seismic station if the earth is approximated to be made of isotropic homogeneous material with density
ρ = 3000Kg/m3 Youngs modulus Y = 5×1010 N/m2 and Poisons
ratio ρ = 0.2.
2. A solid body made of homogeneous isotropic material with shear modulus µ and Lame’s constant λ. The undeformed shape of this body
is a cube with edges of length L. Under the action of some external
forces, it comes to static equilibrium. In a coordinate system with the
origin at the center of the cube and the axes parallel to the edges, the
deformed shape is described by the displacement vector field,
~u =
1 2 L2
(x −
)yb
R
4
(a) Sketch the shape of the deformed cube.
(b) Compute all the components of the stress tensor.
(c) Compute the body force density on all the six faces of the cube.
(d) Compute the surface force density on all the six faces of the cube
i.e. x = ±L/2, y = ±L/2 and the z = ±L/2 faces.
1
3. (a) In a wind tunnel experiment the pressure on a sphere of radius R
was measured to be P (θ, φ) when the wind velocity at points far
away from the spheres was ~u, (θ, φ being the polar coordinates of
the sphere). If the experiment was to be repeated for a sphere
of radius 2R and wind velocity 21 ~u, what would be the measured
pressure?
(b) An inviscid, incompressible liquid of density ρ is flowing through
a pipe of cross section area A with speed v. The pressure in the
pipe at the inlet is P . This pipe splits into two pipes, both having
cross section A/2. The pressure at the outlet of these two pipes
is maintained at P1 and P2 . Assuming steady potential flow, and
no gravitational potential changes, compute the speeds v1 and v2
of the liquid at the outlets of the two outgoing pipes.
4. An incompressible liquid with the coefficient of viscosity, η and density ρ is
flowing in a pipe of radius R and length L. There is a pressure difference of
∆P between the ends of the pipe. Compute the total mass of fluid passing
through any cross section of the pipe per unit time.
2