Some Applications of Double Integrals in Mechanics and Strength of Materials

The coordinates of the center of gravity of a plate are found by the formulas:

where $\mu =\mu (x,y)$ is the density of the plate at the point $(x,y)$; $M$ is the mass of the plate, and the quantities $Mx_{c}$ and $My_{c}$ are the static moments of the plate relative to the axes $Oy$ and $Ox$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the lines $x=0$, $y=0$, $x+y=a$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the lines $y^{2}=2px$, $y=0$, $x=x_{0}$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{2/3}+y^{2/3}=a^{2/3}$; $x>0$, $y>0$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{2}+y^{2}=a^{2}$; $x>0$, $y>0$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{2}+y^{2}=a^{2}$; $\left|y\right|<\operatorname {tg}\alpha$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $r^{2}=a^{2}\cos 2\varphi$ (the right-hand loop).

Find the coordinates of the center of gravity of a homogeneous plate bounded by the cardioid $r=a(1+\cos \varphi )$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the loop of the curve $x^{3}+y^{3}=3axy$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the lines $x=a(t-\sin t)$, $y=a(1-\cos t)$; $0<t<2\pi$; $y=0$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{4}+y^{4}=x^{2}y$ (the right-hand loop).

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $\sqrt{x}+\sqrt{y}=\sqrt{a}$, $x=0$, $y=0$.

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {xy}{ab}$ (the right-hand loop).

Prove that the volume of a right cylinder (or prism) cut off by a plane not necessarily parallel to the base equals the area of the base multiplied by the height above the center of gravity of the base.

Prove that the volume of the solid obtained by rotating an area $S$ about an axis that does not intersect $S$ and lies in the same plane equals the area $S$ multiplied by the length of the circle traced by the center of gravity of the area $S$. (Pappus's theorem, often called Guldin's theorem.)

The moment of inertia of an area $S$ relative to some axis lying in the same plane is called the integral $\displaystyle J=\iint \limits_{S} \delta^{2} \, dx \, dy$, where $\delta$ is the distance from the variable point $(x,y)$ to the axis, and the integral is taken over the whole area $S$. In particular, the moments of inertia relative to the axes $Ox$ and $Oy$ equal the integrals $J_{x}=\iint \limits_{S} y^{2} \, dx \, dy$ and $J_{y}=\iint \limits_{S} x^{2} \, dx \, dy$. The polar moment of the area $S$ relative to some point is called the integral $\iint \limits_{S} r^{2} \, dx \, dy$, where $r$ is the distance from the point $(x,y)$ to the given point. In particular, the polar moment relative to the origin equals $\iint \limits_{S}(x^{2}+y^{2}) \, dx \, dy$. The centrifugal moment is called the integral $K_{xy}=\iint \limits_{S} xy \, dx \, dy$. The curve $J_{x}x^{2}-2K_{xy}xy+J_{y}y^{2}=1$ is called the ellipse of inertia of the area $S$.

Find the moments of inertia of a sector with radius $a$ and central angle $2\varphi$ relative to its axis of symmetry and relative to the perpendicular to it passing through the center of gravity.

Find the moment of inertia of a segment with radius $a$ and central angle $2\varphi$ relative to its axis of symmetry.

Can the ellipse of inertia of a sector become a circle?

An isosceles triangle has base $a$ and height $h$. For what relation between $a$ and $h$ does the ellipse of inertia of the triangle relative to the center of gravity become a circle?

Find the moments of inertia of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ relative to the axes, and find the equation of the ellipse of inertia.

A ring between concentric circles of radii 13 and 12 has the same area as a circle of radius 5. How many times greater is the moment of inertia of the ring than that of the circle, if both moments are taken relative to a diameter?

Find the moment of inertia relative to the axis $Ox$ of the parallelogram with sides $a_{1}x+b_{1}y=\pm h_{1}$, $a_{2}x+b_{2}y=\pm h_{2}$.

Find the centrifugal moment of inertia of the same parallelogram (with sides $a_{1}x+b_{1}y=\pm h_{1}$, $a_{2}x+b_{2}y=\pm h_{2}$).

Find the moment of inertia of the area of the curve $x^{4}+y^{4}=x^{2}+y^{2}$ relative to the axis $Ox$ and compare its value with the same moment of inertia of the square $\left|x+y\right|+\left|x-y\right|=2$.

The torsional rigidity of a rod with a simply connected cross-section $S$ is measured by the quantity $C=2\mu \iint \limits_{S} \psi (x,y) \, dx \, dy$, where $\mu$ is the shear modulus of the rod's material, and $\psi (x,y)$ is the stress function. The function $\psi (x,y)$ equals zero on the boundary of the cross-section, and inside it satisfies the equation $\dfrac {\partial^{2}\psi }{\partial x^{2}}+\dfrac {\partial^{2}\psi }{\partial y^{2}}=-2$.

Find the rigidity $C$ for a circular cross-section, when $2\psi =a^{2}-x^{2}-y^{2}$.

Find $C$ for an elliptical cross-section, when $(a^{2}+b^{2})\psi =a^{2}b^{2}-b^{2}x^{2}-a^{2}y^{2}$, where $a$ and $b$ are the semi-axes of the ellipse.

The same question (find the torsional rigidity $C$) as in problem 3658, for an equilateral triangle with sides $x\pm y\sqrt{3}=h$, $x=0$. In this case $2h\psi =x(h-x-y\sqrt{3})(h-x+y\sqrt{3})$.

The level of a liquid is the axis $Ox$, and the axis $Oy$ is directed downward. Prove that the pressure of the liquid on a vertical plate $S$ equals $\gamma \iint \limits_{S} y \, dx \, dy$, where $\gamma$ is the specific weight of the liquid. Prove that the point of application of the resultant of the pressure forces lies at a depth $h$, where $\displaystyle h\iint \limits_{S} y \, dx \, dy=\iint \limits_{S} y^{2} \, dx \, dy$.