Section 1
Solving differential equations
The pressure, $p$, on an object under a fluid of density $\rho$ (Greek letter rho) is given by$$\frac{\mathrm{d} p}{\mathrm{~d} z}=-\rho g$$where $z$ represents depth and $g$ is the acceleration due to gravity. Find an expression for $p$.
Consider a fluid rotating about a vertical axis. The pressure $p$ at radius $r$ from the axis is given by$$\frac{\mathrm{d} p}{\mathrm{~d} r}=-\rho \omega^{2} r$$where $\rho$ is the density and $\omega$ is the angular velocity of the fluid. Find an expression for $p$.
The streamlines of a fluid flow are given by$$\frac{\mathrm{d} y}{\mathrm{~d} x}=C$$where $C$ is a constant. Plot the streamline that passes through the origin and $C=5$.
The streamlines of a fluid flow are given by$$\frac{\mathrm{d} y}{\mathrm{~d} x}=e^{x}$$Solve the differential equation and sketch the streamlines.
The acceleration, $a$, of an object is defined as$$a=\frac{\mathrm{d} v}{\mathrm{~d} t}$$ where $v$ is the velocity of the object at time $t$. Given that when $t=0, v=u$, and the acceleration, $a$, is constant, show that $v=u+a t$.
The acceleration, $a$, of a particle is given by$$a=5-3 t$$Given that the initial displacement at $t=0$ is $-2.1 \mathrm{~m}$ and the initial velocity is $8 \mathrm{~m} / \mathrm{s}$, find expressions for the velocity and displacement.[Hint: We have, $a=\frac{\mathrm{d} v}{\mathrm{~d} t}, v=\frac{\mathrm{d} s}{\mathrm{~d} t}$ where $s$ is the displacement
An object rotates with angular acceleration, $\alpha$, defined as$$\alpha=\frac{\mathrm{d} \omega}{\mathrm{d} t}$$where $\omega$ is the angular velocity. Given that when $t=0, \omega=\omega_{0}$, show that$$\omega=\omega_{0}+\alpha t$$Also the angular velocity $\omega$ is defined as$$\omega=\frac{\mathrm{d} \theta}{\mathrm{d} t}$$where $\theta$ is the angular displacement. Given that when $t=0, \theta=0$ show that$$\theta=\omega_{0} t+\frac{1}{2} \alpha t^{2}$$
The streamlines of a fluid flow are given by$$\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{x}{y}$$ Show that $x^{2}+y^{2}=A$ where $A$ is a constant. Sketch the streamlines for $A=1,25$ and 100
The streamlines of a fluid flow are given by$$\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{y}{x}(y>0 \text { and } x>0)$$Show that $y=\frac{A}{X}$ and sketch on the same axes the streamlines for $A=1$, 5 and 8 ( $A$ is a constant).
The streamlines of a fluid flow are given by the first order differential equation$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y+1}{x+1}(y>-1$ and $x>-1)$Show that $y=A x+A-1$ where $A$ is a constant.
The temperature gradient, $\frac{\mathrm{d} \theta}{\mathrm{d} x}$, of a slab of thickness $t$ and with thermal conductivity $k$ is given by$$\frac{\mathrm{d} \theta}{\mathrm{d} x}=C$$where $C$ is a constant and $\theta$ is a function of $x$.By using the conditions $\theta(0)=\theta_{1}, \theta(t)=\theta_{2}$ and Fourier's law$$Q=-k A \frac{\mathrm{d} \theta}{\mathrm{d} x}$$
The pressure, $p>0$, of a gas in isothermal condition is given by$$\frac{\mathrm{d} p}{\mathrm{~d} z}=-\frac{p m g}{R T}$$where $z$ is the altitude, $T$ (constant) is the temperature, $m$ is the molar mass and $R$ is a gas constant.Show that$$p=A e^{-\frac{m g}{K T} z}$$(where $A$ is a constant).
The pressure, $p$, of the atmosphere at an altitude $z$ is given by$$\frac{\mathrm{d} p}{\mathrm{~d} z}=-k p^{\frac{1}{\gamma}}(k \neq 0)$$where $\gamma$ is the specific heat constant $(\gamma>1)$ and $k$ is a constant. Show that$$z=\frac{\gamma p^{\frac{\gamma-1}{\gamma}}}{k(1-\gamma)}+A$$(where $A$ is a constant).