Differential-equation models of physical processes

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 $\gamma$.

A vessel contains $M$ cm$^{3}$ of solution. Into it, every second, $q$ cm$^{3}$ of water flows continuously and is immediately mixed in. From the vessel, $q$ cm$^{3}$ of solution flows out per second. Prove that the amount of dissolved substance $m$ is expressed by the formula $m=m_{0}e^{-\frac{q}{M}t}$, where $m_{0}$ is the initial amount of substance, and $t$ is the time in seconds from the start of the process.

A rope is wound around a post of radius $r$. The force pressing its element $\Delta s$ against the post is equivalent to the quantity $\frac{T}{r}\Delta s$, where $T$ is the tension of the rope at the given point. Owing to friction, the element can only slip if the force driving it is greater than $kf$, where $k$ is the coefficient of friction and $f$ is the force pressing the element against the post. Derive from this the differential equation $dT=-\frac{kT}{r}\, ds$ and its general integral $T=T_{0}e^{-\frac{ks}{r}}$, where $T$ is the tension at which slipping begins and $T_{0}$ is the tension at the initial point.

Light of brightness $f_{0}$ falls on a light-absorbing medium. Assuming that the absorption by a layer $\Delta x$ is approximately proportional to $\Delta x$ and to the brightness $f$, prove that at depth $x$ the brightness is expressed by the equality $f=f_{0}e^{-kx}$, where $k$ is the coefficient of proportionality.

Rays from a light source are absorbed by the surrounding medium. Assuming that the absorption of light between spheres of radii $r$ and $r+\Delta r$ centered at the light source is, up to small quantities of higher order, equal to $kf\cdot 4\pi r^{2}\Delta r$, prove that the brightness at a distance $r$ from the light source equals $f_{0}\dfrac {e^{-kr}}{r^{2}}$, where $f_{0}e^{-k}$ is the brightness of light at $r=1$, and $k$ is the coefficient of proportionality.

Find the curve that would be equally illuminated everywhere (with a given illuminance $a$) by a point source of light of intensity $A$. It is known that illuminance is inversely proportional to the square of the distance and proportional to the sine of the angle between the ray and the illuminated surface.

Water flows uniformly ($V$ cm$^{3}$/sec) into a cylindrical vessel with base area $S$ cm$^{2}$ and height $H$ cm, and simultaneously partly flows out through an opening in the bottom of area $\sigma$ cm$^{2}$. After what time will the vessel fill to the top, and what is the condition for it to ever fill up? The outflow velocity is determined by Torricelli's law (see problem 3302).

A heavy material point of mass $m$ begins to fall in a resisting medium, with the resistance force depending on the velocity according to the law $f=kv^{n}$. Determine the law of change of velocity over time and the distance traveled in time $t$ for $n=1$; $2$.

A gas of initial density $\rho_{0}$ and pressure $p_{0}$ is contained in a vessel rotating about its axis with constant angular velocity $\omega$. Find the law of pressure distribution in the steady-state motion, when all particles of the gas rotate with angular velocity $\omega$. The gas obeys Boyle-Mariotte's law; neglect the weight of the gas.

If $R$ is the resistance, $L$ is the self-induction coefficient, and $V$ is the voltage, then the current strength in the circuit $I$ satisfies the equation $L\dfrac {dI}{dt}+RI=V$. Find the current strength $t$ seconds after switching on, if $V$ is constant and at $t=0$ the current strength was $I_{0}$.

The same question as in the previous problem (find the current strength $t$ seconds after switching on), for $V=V_{0}\sin 2\pi nt$.

On cooling, the temperature of a body surrounded by a medium of constant temperature $\Theta_{\text{cp}}$ varies with time according to the law $d \Theta / d t = -k\left(\Theta -\Theta_{\text{cp}}\right), k > 0$ (Newton).

Find the dependence of the temperature of a body, having initial temperature $\Theta_{0}$, on time.

A body is surrounded by a medium of constant temperature $\Theta_{\mathrm{cp}} = 20{ }^{\circ } \mathrm{C}$. In 20 minutes the temperature of the body, as a result of cooling, dropped from $100{ }^{\circ } \mathrm{C}$ to $60{ }^{\circ } \mathrm{C}$. In what time from the start of cooling will the temperature of the body drop to $30{ }^{\circ } \mathrm{C}$?

An open pool, of volume $V \mathrm{~ m}^{3}$, is filled with a mixture of two liquids $A$ and $B$ with $m \%$ of liquid $B$ (by volume). Into the pool a mixture of the same liquids, containing $n \%$ of liquid $B$, begins to be fed at a rate of $p \mathrm{~ m}^{3}$/min. Assuming that mixing in the pool occurs instantaneously, find the dependence on time of the percentage content of liquid $B$ in the volume of the pool.

The rate of radioactive decay is proportional to the amount of substance present, with coefficient $\lambda$, the decay constant. In a year $n \%$ of the initial amount of substance decayed. Find its half-life (the time in which half the substance decays).

Suppose that in a chemical reaction, from a mole of substance $A$ and a mole of substance $B$, a mole of substance $C$ is obtained. Let $a$ and $b$ be the initial numbers of moles of substances $A$ and $B, a > b$. Assuming that the rate of the reaction at each moment of time is proportional, with coefficient $k$, to the product of the masses of the reacting substances present at that moment, find the number of moles of substance $C$ by the moment $t > 0$.

The rate of a chemical reaction converting substance $A$ into substance $B$ is proportional to the product of the concentrations of these substances. The initial concentration of substance $B$ equals $\mu_{0}$ (the ratio of the amount of $B$ to the total amount of substances $A$ and $B$). Find the dependence of the concentration of substance $B$ on time.

Under the conditions of the previous problem, find the concentration of substance $B$ after $t = 1$ h, if $\mu_{0} = 0,2$, and the concentration after $t = 1 / 4$ h equals 0.8.

A current $I$ in a circuit with resistance $R$ and coefficient of self-induction $L$ satisfies the equation

where $V$ is the external voltage applied to the circuit. Find the dependence $I(t)$ of the current in the circuit on time, if:

A bullet, having pierced a board of thickness $h$, changed its velocity from the value $v_{1}$ to the value $v_{2}$. Assuming the resistance force is proportional to the square of the velocity, find the time of motion of the bullet in the board.

A material point located on the surface of the Earth (radius of the Earth equals $R$) is given an initial vertical velocity $v_{0} = \sqrt{2 g R}$ (the second cosmic velocity). In what time will the point move away from the surface of the Earth to a distance equal to $3 R$ (neglecting air resistance)?

A flywheel, immersed in a liquid, is set into rotation from rest by a driving torque $M$. The moment of resistance of the liquid varies according to the law $M_{c} = k \omega^{2}$, where $k$ is a constant, $\omega$ is the angular velocity of the flywheel. In what time will the angular velocity of the flywheel increase to the value $\omega_{0}$?

A homogeneous rod of length $2 l$, lying on a horizontal plane, is given angular velocity $\omega$ around a vertical axis passing through the center of the rod. Determine the time of rotation of the rod until it stops, assuming its pressure on the plane is constant along the length of the rod, and the coefficient of friction with the plane equals $k$.

A point, having initial mass $m_{0}$ and velocity $v_{0}$, moves rectilinearly. The velocity, relative to the point, of the particles separating from it is constant and equals $u$; there are no external forces. Find the mass of the point at the moment when its velocity equals $v_{1}$.

The mass of a rocket with fuel equals $M$, without fuel $-m$, the exhaust velocity of the combustion products relative to the rocket is constant and equals $u$, the initial velocity of the rocket equals zero. Find the velocity of the rocket after the fuel has burned out.

A rocket, having initial velocity $v_{0}$ and mass $m_{0}$, is decelerated by its engine to zero velocity. The fuel consumption per unit time is constant and equals $\mu$, the exhaust velocity of the combustion products is constant and equals $u$. Find the length of the braking path.

A rocket moves vertically upward from the surface of the Earth, with initial velocity $v_{0}$. The exhaust velocity of the combustion gases relative to the rocket is constant and equals $u$. It is required to reach velocity $v_{1}$ in time $T$, with final mass $m_{1}$. What must the mass of the fuel be (neglecting the change of gravity, air resistance, and the rotation of the Earth)?

A rocket starts from the surface of the Earth with initial mass $m_{0}$ and moves vertically upward with constant velocity $v_{0}$. The exhaust velocity of the combustion products relative to the rocket is constant and equals $u$. Find the dependence of the mass of the rocket on time. The radius of the Earth equals $R$; neglect air resistance and the rotation of the Earth.

A drop of water falls in still air under the action of constant gravity (neglecting air resistance). The initial mass, radius and velocity of the drop equal respectively $m_{0}, r_{0}$ and $v_{0}$. Due to condensation of vapor, the mass of the drop increases. Find the velocity of the drop at the moment when its radius has increased by a factor of $\alpha$, if the rate of increase of the drop's mass varies with the radius $r$ according to the law:

A cylinder of height $H$ and radius $R$ is filled with liquid. The axis of the cylinder is vertical. In the bottom of the cylinder a small opening of area $S$ is open. In what time will the liquid flow out of the cylinder?

A cylindrical tank, positioned vertically, has a small opening in the bottom. Half the water from the full tank flowed out in $t$ minutes. In what time will all the water flow out?

A conical vessel with vertical axis, base radius $R$, and height $H$ is filled with liquid. At the vertex, located at the bottom, a small opening of area $S$ is open. In what time will the liquid flow out of the vessel?

What shape must a vessel that is a solid of revolution have so that the lowering of the liquid level, as it flows out of a small opening in the lower part, is uniform?

The resistance of an electric circuit equals $R$, the coefficient of self-induction $-L$. Find the amount of heat released on the resistance $R$ when:

A mass $m$ of liquid with specific heat capacity $c$ is in contact with a medium whose temperature is constant and equals $\Theta_{0} > 0$. Due to heat dissipation into the surrounding medium the liquid may cool. It is known that the cooling process obeys Newton's law (problem 114), and that cooling this mass of liquid from temperature $3 \Theta_{0}$ to temperature $2 \Theta_{0}$ takes time $\tau$. The liquid begins to be heated by heat released on a constant resistance $R$, to which a constant voltage $V$ is applied. In what time will the liquid heat up from temperature $\Theta_{0}$ to temperature $4 \Theta_{0}$?

An axisymmetric rod of length $l$ is fixed at one end, while a force $P$ is applied along the axis at the other. The specific weight of the rod's material equals $\gamma$, the modulus of longitudinal elasticity $E$. Find the dependence of the cross-sectional area $S(x)$ of the rod on the distance $x$ from its fixed end, from the condition of constancy of the normal stress in all cross-sections, equal to $\sigma_{0}$ (a rod of uniform strength). Solve the problem under the condition that:

A rope wraps around a fixed cylindrical drum along an arc of angular size $\varphi_{0}$. Suppose the model of "dry" friction holds, i.e. at every point of contact of the rope with the drum the maximum value of the tangential stress is proportional to the normal stress with coefficient of friction $\mu =$ const. Suppose the tension of the rope on one side equals $T_{1}$. Find the greatest tension of the rope on the other side for which the rope will still not start to slip (Euler).