Problems in mechanics and physics

Form appropriate integral sums and find their limits to solve the following problem: the length of a rod is $l=10$ meters; if the linear density of the rod varies according to the law $\delta =6+0.3x$ kilograms/meter, where $x$ is the distance from one of the two endpoints of the rod, find the mass of the rod.

Form appropriate integral sums and find their limits to solve the following problem: a body of mass $m$ is raised from the surface of the Earth (whose radius is $R$) to a height $h$. How much work must be done? If the body is moved off to infinity, what does the work equal?

Form appropriate integral sums and find their limits to solve the following problem: if a force of 1 kilogram stretches a spring by 1 centimeter, how much work must be done to stretch this spring by 10 centimeters?

Form appropriate integral sums and find their limits to solve the following problem: a cylinder of diameter 20 centimeters and length 80 centimeters is filled with steam at a pressure of 10 kilograms/centimeter${}^{2}$. Assuming the temperature of the gas remains unchanged, how much work must be done to reduce the volume of the gas by half?

Form appropriate integral sums and find their limits to solve the following problem: find the pressure of water on a vertical wall shaped like a semicircle of radius $a$, whose diameter lies on the surface of the water.

Form appropriate integral sums and find their limits to solve the following problem: find the pressure of water on a vertical wall shaped like a trapezoid, whose lower base is $a=10$ meters, upper base $b=6$ meters, height $h=5$ meters, with the lower base submerged at a depth of $c=20$ meters below the water surface.

Form differential equations to solve the following problem: the velocity of a moving point varies according to the following law:

What is the distance traveled by this point over the closed interval $(0,T)$?

Form differential equations to solve the following problem: a uniform sphere of radius $R$ and density $\delta$ rotates about its diameter with angular velocity $\omega$. Find the kinetic energy of this sphere.

Form differential equations to solve the following problem: with what force does an infinite straight line of constant linear density $\mu_{0}$ attract a particle of mass $m$ at distance $a$ from the line?

Form differential equations to solve the following problem: compute the force with which a circular thin plate of radius $a$ and constant surface density $\delta_{0}$ attracts a particle $P$ of mass $m$, where the particle lies on the perpendicular line through the center $Q$ of the plate and perpendicular to the plane of the plate, with the shortest distance $PQ$ equal to $b$.

Form differential equations to solve the following problem: by Torricelli's law, the velocity at which liquid flows out of a container equals

where $g$ is the acceleration due to gravity, $h$ is the height of the liquid surface above the opening, and $c=0.6$ is an experimental coefficient.

An upright cylindrical tank of diameter $D=1$ meter and height $H=2$ meters, once filled, drains through a circular hole of diameter $d=1$ centimeter in its base. How long does it take to empty completely?

Set up a differential equation to solve the following problem: what shape should the vessel of a solid of revolution have so that, as the liquid flows out, the level of the liquid surface drops uniformly?

Set up a differential equation to solve the following problem: the rate of decay of radium at each instant is proportional to the amount currently present. Suppose that at the initial instant $t=0$ there is $Q_{0}$ grams of radium, and after a time $T=1600$ years the amount has decreased by half. Find the law of decay of radium.

Set up a differential equation to solve the following problem: the rate of a second-order chemical reaction converting substance $A$ into substance $B$ is proportional to the product of the concentrations of the two substances. What is the percentage of substance $B$ contained in the vessel after $t=1$ hour, given that at $t=0$ minutes there is $20\%$ of substance $B$, and at $t=15$ minutes it has become $80\%$?

Set up a differential equation to solve the following problem: by Hooke's law, the relative elongation $\varepsilon$ of a rod is proportional to the stress $\sigma$ on the corresponding cross-section, that is,

where $E$ is Young's modulus.

Find the elongation of a heavy conical rod, with the cone's apex pointing downward and the base fixed, given that the base radius equals $R$, the height of the cone is $H$, and the specific weight is $\gamma$.