RGU Question Paper Mathematics Semester VI 2019: Hydrostatics and Mathematical Methods

June-2019

B.A/B. Sc. (VI Semester) Examination

MATHEMATICS

Paper: MATH-362

(Hydrostatics and Mathematical Methods)

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Notes: (i) Answer all questions.

(ii) The figures in the margin indicate full marks for the questions.

I. Answer any five parts: 𝟐 × 𝟓 = 𝟏𝟎

(a) Show that the positions of equilibrium of a body floating in homogeneous liquid are determined by drawing normals from $G$, the center of mass of the body, to the surface of buoyancy.

(b)  Express $x^4+2x^3+2x^2-3$ in terms of Legendre's polynomial.

(c) State the principle of Archimedes. What is centre of buoyancy?

(d) Express $J_2\left(x\right)$ in terms of $J_0\left(x\right)$ and $J_1\left(x\right)$.

(e) Find Express $x^4+2x^3-5x^2-x-2$ in terms of Legendre polynomial.

(f) Find the Laplace transform of $\mathcal{L}\left(t^3{e}^{5t}\right)$.

(g) If $\mathcal{F}\left\{F\left(x\right)\right\}=f\left(s\right)$, then show that $\mathcal{F}\left\{F(x-a)\right\}=e^{-ias}f\left(s\right)$.

(h) What do you mean by a surface of equal pressure? Define centre of pressure of a plane area immersed in a fluid. 

II. Answer any four parts: 𝟓 × 𝟒 = 𝟐𝟎

(a) If a mass of fluid is at rest under the action of given forces, obtain the equation which determines the pressure at any point of the fluid. 

(b) Show that

(c) Show that the equilibrium is stable or unstable according as the meta-centre is above or below the centre of gravity of the body.

(d) If 𝐿{𝑓(𝑡)} = 𝐹(𝑠), then show that

provided the integral exists.

(e) Prove that

(f) If the floating solid be a cylinder, with its axis vertical, the ratio of whose specific gravity to that of the fluid is $\sigma$, prove that the equilibrium will be stable, if the ratio of the radius of the base to the height be greater than ${\left[2\sigma \left(1-\sigma \right)\right]}^{1/2}$.

(g) Prove that when $n$ is a positive integer

(h) A hollow vessel containing some liquid, floats in a liquid. Determine the nature of equilibrium supposing that the body is symmetrical with respect to the vertical plane of displacement through its centre of mass and that the centre of mass of the body and that of the liquid contained are in the same vertical line.

III. Answer any five parts: 𝟏𝟎 × 𝟓 = 𝟓𝟎

(a)(i) Show that the position of the centre of pressure relative to the area remains unaltered by rotation about its line of intersection with the effective surface. (5)

(ii) If the components parallel to the axes of the forces acting on the element of fluid at $(x,y,z)$   be proportional to $y^2+2\lambda yz+z^2$, $z^2+2\mu zx+x^2$, $x^2+2\gamma xy+y^2$, show that if equilibrium is possible, $2\lambda =2\mu =2\gamma =1$. (5)

(b)(i) Show that: (5)

(ii) Prove that: (5)

(c)(i) A liquid of given volume V is at rest under the forces

Find the pressure at any point of the liquid and the surfaces of equal pressure. (5)

(ii) Show that the forces represented by

will keep a mass of liquid at rest, if the

from the plane x + y + z = 0, and the curves of equal pressure and density will be circles. (5)

(d)(i) Show that : (6)

(ii) Evaluate: (4)

(e)(i) Show that the depth of the centre of pressure of the area included between the arc and the asymptote of the curve $(r-a)\cos \left(\theta \right)=b$ is

the asymptote being in the surface and the plane of the curve is vertical. (7)

(ii)  If the meta-centre is above the centre of gravity, what type of equilibrium can you expect? State the principle of Archimedes. (3)

(f)(i) Find the inverse Laplace transform of (5)

(ii) Find: (5)

(g)(i) Find the Fourier cosine transform of the function (5)

(ii) Find the Fourier transform of (5)

and hence evaluate

(h) (i)Using Laplace transform, solve : (5)

(ii) Solve: (5)