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

 

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BSc (VI Semester) Examination

MATHEMATICS

Paper : MATH–361 (New 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 $P{\text{'}}_{2n}\left(0\right)=0$.

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

(c) A rectangular lamina of width $b$ and depth $d$ is submerged vertically in water, such that the upper edge of the lamina is at a depth $h$ from the free surface. Show that the depth of the centre of pressure is
$h+\frac{d}{3}\frac{3h+2d}{2h+d}$

(d) If $f\left(s\right)$ is the Fourier transform of $F\left(x\right)$, prove that the Fourier transform of
$F\left(ax\right)$ is $\frac{1}{a}f\left(\frac{s}{a}\right)$.

(e) Show that the surface of equal pressure is intersected orthogonally by the lines of force.

(f) If $f\left(s\right)=\mathcal{L}\left(F\right(t\left)\right)$ denotes the Laplace transform of the function $F\left(t\right)$, prove
that ${\mathcal{L}}^{-1}\left\{f\right(as\left)\right\}=\frac{1}{a}f\left(\frac{t}{a}\right), a>0$

(g) Obtain the differential equations of the lines of forces at any point
$(x,y,z)$.

(h) Show that the differential equation
$x^{2}y\text{'}\text{'}+3xy\text{'}+(1+x)y=0$
has a regular singular point at $x=0$.

(i) Find the whole pressure on a triangle, the depths of whose vertices are $h_1$, $h_2$ and $h_3$, the liquid being homogeneous.

2. 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) Find the Laplace transform of $H(t)$defined as
$H\left(t\right)=\left\{\begin{array}{ll}t+1& 0\le t\le 2\\ 3& t>2\end{array}\right.$

(c) Find the sine Fourier transform of the function
$F\left(x\right)=\left\{\begin{array}{ll}\sin \left(x\right)& 0\le x\le a\\ 0& x>a\end{array}\right.$

(d) 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.

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

(f) Prove that
$J_{1/2}=\sqrt{(2/\pi x)}\sin x$

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

3. Answer any five parts : 10×5=50

(a) Derive the solution of Bessel’s equation of $n^{th}$ order by Frobenius method. 10

(b) (i) A lamina in the shape of a quadrilateral $ABCD$ has the side $CD$ in the surface and the sides $AD$, $BC$ vertical and of lengths $\alpha$ and $\beta$ respectively. Prove that depth of the centre of pressure is
$\frac{1}{2}\frac{{\alpha }^4-{\beta }^4}{{\alpha }^3-{\beta }^3} or \frac{1}{2}\frac{(\alpha +\beta )({\alpha }^2+{\beta }^2)}{{\alpha }^2+\alpha \beta +{\beta }^{2.}}$ 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_n\left(x\right)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}[x\pm \sqrt{\left(x^2-1\right)}\cos \varphi ]d\mathrm{\varphi . }$5

(ii) Verify directly that the representation
$J_0\left(x\right)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\cos \left(xsin\theta \right)d\theta$
satisfies Bessel’s equation in which $n=0$. 5

(d) (i) A vessel in the form of an elliptic paraboloid, whose axis is vertical and equation
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{h}$
is divided into four equal compartments by its principal planes. Into one of these, water is poured to the depth $h$. Prove that, if the resultant pressure on the curved portion be reduced to two forces, one vertical and the other horizontal, the line of
action of the later will pass through the point $(\frac{5}{16}a,\frac{5}{16}b,\frac{3}{7}h).$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, we must have $2\lambda =2\mu =2\gamma =1.$  5

(e) (i) Show that
$\underset{0}{\overset{\pi }{\int }}{P}_n\left(cos\theta \right)cos\left(n\theta \right)d\theta =\frac{1.2.3...(2n-1)}{2.4.6...2n}\pi =B(n+\frac{1}{2},\frac{1}{2}). \;$5

(ii) A mass of fluid rests upon a plane subject to a central attractive force $\frac{\mu }{r^2}$, situated at a distance $c$ from the plane on the side opposite to that on which is the fluid; show that the pressure on the plane is
$\frac{\pi \rho \mu (a-c{)}^2}{a}$
$a$ being the radius of the sphere of which the fluid, on the plane in the form of a cap, is a part. 5

(f) (i) Using Laplace transform method, solve
$y\text{'}\text{'}\left(t\right)+y\left(t\right)=t$ given that $y\text{'}\left(0\right)=1, y\left(\pi \right)=0.$  5

(ii) Using suitable Fourier transform, solve
$\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}$
if $u(0,t)=0, u(x,0)=e^{-x}, u(x,t)$is bounded. 5

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

(ii) Show that the specific gravity of a mixture of $n$ liquids is greater when equal volumes are taken than when equal weights are taken; assuming no change in
volume as the result of mixing. 10