RGU Question Paper Mathematics Semester VI 2020: Hydrostatics and Mathematical Methods(Old)

 

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BSc (VI Semester) Examination

MATHEMATICS

Paper : MATH–362 (Old Course)

( Hydrostatics and Mathematical Methods )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five parts : 2×5=10

(a) Prove that $P_n\left(1\right)=1$

(b) Find the Laplace transform of $e^{-t}$ .

(c) Show that the homogeneous liquid will be in equilibrium only when the
system of forces is conservative.

(d) If $f\left(s\right)$ is the Fourier transform of $F\left(x\right)$, prove that the Fourier transform of
$F\text{'}\left(x\right)$, the derivative of $F\left(x\right)$, is $\left\{isf\right(s\left)\right\}$.

(e) State the principle of Archimedes. What is the centre of pressure?

(f) If ${\mathcal{L}}^{-1}\left\{\frac{s^2-1}{{\left(s^2+1\right)}^2}\right\}=t\cos \left(t\right)$, prove that
${\mathcal{L}}^{-1}\left\{\frac{9s^2-1}{{\left(9s^2+1\right)}^2}\right\}=\frac{t}{9}\cos \left(\frac{t}{3}\right)$

(g) Show that
$x^3=(2/5)P_3\left(x\right)+(3/5)P_2\left(x\right)$. 

Answer any four parts : 5×4=20

(a) Determine the necessary condition that must be satisfied by a given distribution of forces $X,Y,Z$ so that the fluid may maintain equilibrium.

(b) If $f\left(t\right)$ be a periodic function of period $a$, show that
$\mathcal{L}\left\{f\right(t\left)\right\}=\frac{1}{1-e^{-pa}}\underset{0}{\overset{a}{\int }}{e}^{-st}f\left(t\right)dt$

(c) Find the sine Fourier transform of the function
$F\left(x\right)=\frac{1}{x}{e}^{-ax}$

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

(e) If $P_0\left(\cos \left(\theta \right)\right)=1$, show that
$1+\frac{1}{2}{P}_1\cos \left(\theta \right)+\frac{1}{3}{P}_2\left(\cos \left(\theta \right)\right)+...=\log \left(\frac{1+\sin \left(\theta /2\right)}{\sin \left(\theta /2\right)}\right)$

(f) Prove that
$\cos \left(x\sin \left(\varphi \right)\right)=J_0+2J_2\cos \left(2\varphi \right)+2J_4\cos \left(4\varphi \right)+...$

(g) Define meta-centre and metacentric height of a floating body. What do you mean by curves of floatation and curves of buoyancy?

(h) If $n$ is non-negative, prove that
$\underset{0}{\overset{\infty }{\int }}{J}_n\left(bx\right)dx=\frac{1}{b}$

3. Answer any five parts : 10×5=50

(a) Derive the solution of Legendre’s equation by Frobenius method. 10

(b) (i) A right cone is totally immersed in water and the depth of the centre of its base being given. Prove that $P,P\text{'},P\text{'}\text{'}$ being the resultant pressures in its convex
surface when the sines of the inclination of its axis to the horizontal are $s,s\text{'},s\text{'}\text{'}$ respectively
$P^2(s\text{'}-s\text{'}\text{'})+P{\text{'}}^2(s\text{'}\text{'}-s)+P\text{'}{\text{'}}^2(s-s\text{'})=0$ 5

(ii) Prove that if the forces per unit of mass at $(x,y,z)$ parallel to the axes are $y(a-z),x(a-z),xy$, the surfaces of equal pressure are hyperbolic paraboloids and the curves of equal pressure and density are rectangular hyperbolas. 5

(c) (i) Prove that for any +ve integer $n$
$P_{2n+1}\left(0\right)=0, P_{2n}\left(0\right)=(-1{)}^n\frac{\left(2n\right)!}{2^{2n}(n!{)}^{\mathrm{2 \; }}}$5

(ii) Express $J_4\left(x\right)$ in terms of $J_{\mathrm{0 }}$and $J_1$. 5

(d) (i) Express the function

as a Fourier integral. Hence evaluate

(ii) A solid cone is just immersed in water with a generating line in surface. Prove that the inclination to the vertical of the resultant thrust on the curved surface is

where $2\alpha$ is the vertical angle of the cone. 5

(e) (i) Three fluids whose densities are in AP fill a semi-circular tube whose bounding diameter is horizontal. Prove that the depth of one of the common surfaces is double that of the other. 5

(ii) Prove that

5

(f) (i) Using Laplace transform method, solve

5

(ii) Using suitable Fourier transform, solve

if $u(0,t)=u(4,t)=0, u(x,0)=2x$ is bounded. 5

(g) (i) A hemispherical bowl is filled with water and two vertical planes are drawn through its central radius, cutting off a semi-line of the surface. If $2\alpha$ be the angle between the planes, prove that the angle, which the resultant pressure on the surface makes with the vertical, is

5

(ii) Prove that

5

(h) (i) For zeroth-order Bessel’s function, prove that

5

(ii) A solid cone of semi-vertical angle $\alpha$, specific gravity $\sigma$, floats in equilibrium in the liquid of specific gravity $\rho$ with its axis vertical and vertex downwards. Show that the equilibrium is stable, if

5