Answer any ten questions (2 marks each). All questions solved below.
State Stokes' theorem of vectors.
Stokes' Theorem states that the line integral of a vector field A around a closed curve C is equal to the surface integral of the curl of A over any surface S bounded by C. Integral(A.dl) = Integral(Curl A . dS).
Define scalar or dot product of two vectors.
The scalar product of two vectors A and B is defined as the product of their magnitudes and the cosine of the angle theta between them. A.B = |A||B| cos(theta).
Find the area of the parallelogram whose adjacent sides are i-2j+3k and 2i+j-4k.
Area = |A x B|.
State the law of conservation of linear momentum.
If the net external force acting on a system of particles is zero, the total linear momentum of the system remains constant.
Define centre of mass and centre of gravity.
What are the dimensions of moment of inertia?
Moment of Inertia (I) = Mass * (Distance)². Dimensions: [M L² T⁰].
State Newton's law of gravitation.
Every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G(m1m2)/r².
What is a geostationary satellite?
A satellite that appears stationary relative to a point on Earth's surface because it orbits Earth in the equatorial plane with the same angular velocity as Earth's rotation.
What is a central force? Give an example.
A force whose magnitude depends only on the distance from a fixed point and is directed along the line joining the particle to that point. Example: Gravitational force.
State Hooke's law.
Within the elastic limit, the stress applied to a body is directly proportional to the strain produced in it. Stress proportional to Strain.
Calculate the force to double length of steel wire (A=0.5 sq. cm, Y=2x10^11).
To double length, strain = 1. F = Y * A * strain = (2x10^11) * (0.5 x 10^-4 m²) * 1 = 10^7 N.
Difference between angle of twist and angle of shear?
Angle of shear is the relative displacement of the layers of the cylinder, while angle of twist is the angle through which the end of the cylinder is rotated.
Define surface tension of a fluid.
The property of a liquid surface that acts as a stretched elastic membrane, measured as the force per unit length acting perpendicular to an imaginary line drawn on the surface.
Discuss variation of viscosity of a liquid with temperature.
The viscosity of a liquid decreases with an increase in temperature because the intermolecular forces weaken as the kinetic energy of molecules increases.
What is an inertial frame of reference?
A frame of reference in which Newton's first law of motion holds true, meaning the frame is either at rest or moving with a constant velocity.
Answer any five questions (10 marks each). All questions solved below.
(a) Define scalar triple product and show it represents volume of a parallelepiped.
Scalar triple product A.(B x C) is the dot product of one vector with the cross product of the other two.
The cross product (B x C) gives the area of the base, and the dot product with A projects it onto the height, resulting in Base Area * Height = Volume.(b) Solenoidal vector field. Prove A=3y²z²i + 3x²z²j + 3x²y²k is solenoidal.
A vector is solenoidal if its divergence is zero (div A = 0).
(a) First and second-order homogeneous differential equations.
First-order: M(x,y)dx + N(x,y)dy = 0 where M and N are homogeneous functions. Second-order: a(d²y/dx²) + b(dy/dx) + cy = 0.
(b) Solve: (i) dy/dx = x(2 log x + 1) / (sin y + y cos y); (ii) y'' - 6y' + 9y = 0.
(a) Linear momentum from Newton's laws.
(b) Work-energy theorem.Newton's 2nd Law states F = dp/dt. If F_ext = 0, then dp/dt = 0, which means p is constant.
Work-Energy Theorem: The work done by the net force on an object is equal to the change in its kinetic energy.
(a) Angular momentum and torque. Show torque = dl/dt.
Angular momentum L = r x p. Torque tau = r x F. Since dL/dt = d(r x p)/dt = (dr/dt x p) + (r x dp/dt) = (v x mv) + (r x F) = 0 + tau. Thus, tau = dL/dt.
(b) Moment of inertia of thin spherical shell about diameter.
For a thin spherical shell of mass M and radius R, the moment of inertia about any diameter is I = (2/3)MR².
(a) Kepler's laws. Show areal velocity is constant.
Constant areal velocity (dA/dt = L/2m) is a direct consequence of the conservation of angular momentum in a central force field.
(a) Geosynchronous orbit and GPS.
Geosynchronous orbit has a period matching Earth's rotation (24h). GPS uses a constellation of such satellites to triangulate position on Earth using precise timing signals.
(b) Geostationary conditions, applications, and weightlessness.
(a) Define Y, K, n, and sigma.
Young's Modulus (Y), Bulk Modulus (K), Rigidity Modulus (n), and Poisson's ratio (sigma) are elastic constants.
(b) Prove K = Y / [3(1 - 2*sigma)].
This is derived by considering the volumetric strain of a cube under uniform stress in three dimensions and substituting the definitions of Y and sigma.
(a) Couple required to twist a cylinder.
(b) Bending moment and flexural rigidity.For a cylinder of radius R and length L, couple C = (pi * n * R⁴ * theta) / (2L).
Bending moment M = Y*I/R. Flexural rigidity is the product Y*I, representing resistance to bending.(a) Pressure on sides of spherical drop.
(b) Poiseuille's method.Excess pressure P = 2T/R for a drop and 4T/R for a bubble.
Poiseuille's method determines viscosity by measuring the volume of liquid flowing through a capillary tube per unit time: V = (pi * P * r⁴) / (8 * eta * L).(a) Postulates of Special Relativity and Length Contraction.
Length Contraction: L = L0 * sqrt(1 - v²/c²).
(b) Calculate length of 1m rod moving at 0.6c.
L = 1 * sqrt(1 - (0.6)²) = sqrt(1 - 0.36) = sqrt(0.64) = 0.8 m.
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