Riemann surface
matlab

Problem 1.1: Newton’s second law can be expressed as
$$\mathbf{F}=m \mathbf{a}$$
where $\mathbf{F}$ is the net force acting on the body, $m$ mass of the body, and a the acceleration of the body in the direction of the net force. Use Eq. (1) to determine the mathematical model, i.e., governing equation of a free-falling body. Consider only the forces due to gravity and the air resistance. Assume that the air resistance is linearly proportional to the velocity of the falling body.

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Solution: From the free-body-diagram it follows that
$$m \frac{d v}{d t}=F_g-F_d, \quad F_g=m g, \quad F_d=c v$$
where $v$ is the downward velocity $(\mathrm{m} / \mathrm{s})$ of the body, $F_g$ is the downward force ( $\mathrm{N}$ or $\mathrm{kg} \mathrm{m} / \mathrm{s}^2$ ) due to gravity, $F_d$ is the upward drag force, $m$ is the mass (kg) of the body, $g$ the acceleration $\left(\mathrm{m} / \mathrm{s}^2\right)$ due to gravity, and $c$ is the proportionality constant (drag coefficient, $\mathrm{kg} / \mathrm{s})$. The equation of motion is
$$\frac{d v}{d t}+\alpha v=g, \quad \alpha=\frac{c}{m}$$

Problem 1.2: A cylindrical storage tank of diameter $D$ contains a liquid at depth (or head) $h(x, t)$. Liquid is supplied to the tank at a rate of $q_i\left(\mathrm{~m}^3 /\right.$ day) and drained at a rate of $q_0\left(\mathrm{~m}^3 /\right.$ day). Use the principle of conservation of mass to arrive at the governing equation of the flow problem.

Solution: The conservation of mass requires
time rate of change in mass = mass inflow – mass outflow
The above equation for the problem at hand becomes
$$\frac{d}{d t}(\rho A h)=\rho q_i-\rho q_0 \quad \text { or } \quad \frac{d(A h)}{d t}=q_i-q_0$$
where $A$ is the area of cross section of the tank $\left(A=\pi D^2 / 4\right)$ and $\rho$ is the mass density of the liquid.

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