RGU Question Paper Mathematics Semester VI 2021: Hydrostatics and Mathematical Methods

 June-2021

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

MATHEMATICS

Paper: MATH-361 (N/C)

(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) What is hydrostatic equilibrium?

(b) Define fluid. When the fluid will be ideal?

(c) When equal volumes of two substances are mixed the specific gravity (sp. gr.) of mixture is 6, when equal weights of the same substances are mixed, the sp.gr. of the mixture is 4. Find the specific gravities of the substances.

(d) Find the surfaces of equal pressure for a homogeneous fluid acted upon by two forces which vary as the inverse square law of the distance from two fixed points.

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

(f) Find the Laplace transform of

(i) $F\left(t\right)=t^n$ , (ii) 𝐹(𝑡) = sin 𝑎𝑡.

(g) Find the Fourier cosine transform of the function

$F\left(x\right)=\left\{\begin{array}{ll}\cos x& 0<x<a\\ 0& x>a\end{array}\right.$.

(h) Define inverse Fourier transform. Hence, find $F^{-1}\left(\frac{1}{1+s^2}\right)$.

II. Answer any four parts: 𝟓 × 𝟒 = 𝟐𝟎

(a) Show that the necessary and sufficient condition that the component of force (𝑋, 𝑌, 𝑍) per unit mass acting at a particle at a point (𝑥, 𝑦, 𝑧) in fluid may be in equilibrium is $X(\frac{\partial Y}{\partial z}-\frac{\partial Z}{\partial y})+Y\left(\frac{\partial Z}{\partial x}-\frac{\partial X}{\partial z}\right)+Z(\frac{\partial X}{\partial y}-\frac{\partial Y}{\partial x}).$ 

(b) The side 𝐴𝐵 of a triangle 𝐴𝐵𝐶 is in the surface of a fluid and a point 𝐷 is taken in 𝐴𝐶 such that pressure on the triangles 𝐵𝐴𝐷 and 𝐵𝐷𝐶 are equal. Find the ratio 𝐴𝐷:𝐷𝐶.

(c) Find the inverse Laplace transform of

(i)$\frac{64}{81s^4-256}$ 

(ii)$\frac{s}{1+s^2}$

(d) If 𝐿{𝑓(𝑡)} = 𝐹(𝑠), the prove the following:

(i) $L\left\{e^{at}f\right(t\left)\right\}=F(s-a)$

(ii) 𝐿{𝑓(𝑎𝑡)} = (1/𝑎)𝐹(𝑠/𝑎).

(e) If 𝑓(𝑠) is the Fourier transform of 𝐹(𝑥), then prove that $\frac{1}{2}\left\{f\right(s-a)+f(s+a\left)\right\}$ is the Fourier transform of 𝐹(𝑥)cos𝑎𝑥.

(f) Show that $P_n\left(x\right)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}{\left(x^2-1\right)}^2$ where $P_n\left(x\right)$ is a Legendre polynomial of degree 𝑛.

(g) Prove that if the force per unit mass at (𝑥, 𝑦, 𝑧); parallel to the axes are 𝑦(𝑎 − 𝑧), 𝑥(𝑎 − 𝑧), 𝑥𝑦; the surface of equal pressure are hyperbolic hyperboloid and the curves of equal pressure and density are rectangular hyperbolas.

(h) A closed tube in the form of an ellipse with major axis vertical, is filled with liquids of densities ${\rho }_1$, ${\rho }_2$, ${\rho }_3$. $P_1$ is the point of separation of the liquids with densities ${\rho }_2$ and ${\rho }_3$; $P_2$ is the point of separation of the liquids with densities ${\rho }_3$ and ${\rho }_1$, and $P_3$ that of liquids with densities ${\rho }_1$ and ${\rho }_2$. If the distances of $P_1,P_2,P_3$ from the same focus be $r_1,r_2,r_3$, respectively, prove that $r_1({\rho }_2-{\rho }_3)+r_2\left({\rho }_3-{\rho }_1\right)+r_3\left({\rho }_1-{\rho }_2\right)=0$.

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

(a)(i) Find the whole pressure on a plane surface which is in contact with lowest liquids and the fluid consists of layers of different liquids whose densities are ${\rho }_1,{\rho }_2,...,{\rho }_n$ and the thickness $h_1,h_2,...,h_n$, being from the top where the pressure is zero. (5)

(ii) A vessel contains 𝑛 different liquids resting in horizontal layers and of densities ${\rho }_1,{\rho }_2,...,{\rho }_n$ starting from the highest fluid. A triangle is held with its vertex in the nth liquid. Prove that Δ be the area of the triangle and $h_1,h_2,...,h_n$ be the depths of the vertex below the upper surface of the 1st , 2nd , ⋯, nth liquids, respectively, the thrust on the triangle is$\frac{1}{3}\frac{g\Delta }{h_1^2}\left\{{\rho }_1\right(h_1^3-h_2^3)+{\rho }_2\left(h_2^3-h_3^3\right)+...+{\rho }_n{h}_n^3\}$. (5)

(b)(i) Define centre of pressure. Find the centre of pressure of a plane area. (5)

(ii) A circular area of radius 𝑎 is immersed with its plane vertical and centre at a depth ℎ; find the depth of the centre of pressure. (5)

(c)(i) Three fluids are filled up in a semi circular tube whose bounding diameter is horizontal. Prove that the depth of one of the common surface is double that of the other. It is given that the densities of the fluids are in A.P. (5)

(ii) A small uniform tube is bent into the form of a circle whose plane is vertical. Equal volume of two fluids whose densities are 𝜌 and 𝜎 fill half of the tube. Show that radius passing through the common surface makes with the vertical an angle 𝜃, which is given by $\tan \theta =\frac{\rho -\sigma }{\rho +\sigma }$. (5)

(d)(i) Show that resultant vertical thrust on any surface immersed in a liquid is equal to the weight of the super-incumbent liquid and acts vertically downwards through the C. G. of this super-incumbent liquid. (5)

(ii) A solid cone is just immersed with a generating line in the surface, if 𝜃 be the inclination to the vertical of the resultant thrust on the curved surface, prove that $(1-3si{n}^2\alpha )tan\theta =3sin\alpha cos\alpha$, where 2𝛼 being the vertical angle of the cone. (5)

(e)(i) Find the power series solution of Bessel’s differential equation. (5)

(ii) Show that $\frac{d}{dx}{J}_n\left(x\right)=J_{n-1}\left(x\right)-\frac{n}{x}{J}_n\left(x\right)$ and $\frac{d}{dx}\left\{x^n{J}_n\right(x\left)\right\}=x^n{J}_{n-1}\left(x\right)$. (5)

(f)(i) Show that ${\left(1-2xz+z^2\right)}^{1/2}=\sum _{n=0}^{\infty }{z}^n{P}_n\left(x\right), \left|z\right|<1, \left|x\right|\le 1$. (5)

(ii) If $P_n\left(x\right)$ is a Legendre polynomial of degree 𝑛, show that $P{\text{'}}_{n+1}\left(x\right)-P{\text{'}}_{n-1}\left(x\right)=(2n+1)P_n\left(x\right)$. (5)

(g)(i) Use Laplace transform to solve $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-8y=0$; if $y\left(0\right)=3, {\left(\frac{dy}{dt}\right)}_{t=0}=6$. (5)

(ii) Find the Fourier transform of $f\left(x\right)=\left\{\begin{array}{ll}1& \left|x\right|\le a\\ 0& \left|x\right|>a\end{array}\right.$. Hence, evaluate ${\int }_0^{\infty }\frac{\sin ax}{x}dx$.(5)

(h) Using Fourier transform, solve the Laplace equation in a quarter plane $x>0,y>0$.