Fluids

a) (i) [8] A tall cylinder of circular cross sectional area $A$ and height $h$ is filled with a liquid of uniform density $\rho$.

i. Derive an expression for the pressure $P$ at the bottom of the liquid in terms of the variables $\rho, g$ and $h$. Ignore atmospheric pressure.

ii. Show that the units for each side of your derived equation are the same.

iii. What would your result be if the liquid changed in density linearly from $\rho_{t}$ at the top to $\rho_{b}$ at the bottom of the cylinder?

iv. If you used your formula for the pressure at ground level, at the bottom of the Earth's atmosphere, which has a height of about $200 \mathrm{~km}$, it would give an answer which is very far from correct. Explain why.

(ii) A water tank which sits on the ground is kept full of water to a depth of $h$. Water is issuing horizontally from a hole which is in the sidewall of the tank, a distance $h^{\prime}$ below the surface. Determine the maximum horizontal distance $H$ from the hole in the can where the jet of water strikes the floor, by using arguments which assume only

i. conservation of mechanical energy, assuming the water falls down through height $h^{\prime}$,

ii. conservation of momentum (using the pressure at depth $h^{\prime}$ ).

b) [7] An empty, hollow cuboid container of mass $M$ and height $h$, width $a$, depth $b$, is suspended upright from a thin cable in a liquid of density $\rho_{\ell}$, as shown in Figures $3 \mathbf{a}$ and $\mathbf{3 b}$.

(i) Find an expression for the tension, $T$, in the cable by considering the force due to the pressure of the liquid on the top of the container, and the force due to the pressure of the liquid on the bottom of the container.

(ii) The cuboid container now has a small hole pierced in the bottom surface and the liquid leaks in until the air now confined to the top part of the container, occupying a fraction $f$ of the volume, with a fraction $(1-f)$ of the volume inside now being filled with liquid. The pressure of the atmosphere at the surface of the liquid is $P_{0}$ and the density of the liquid is $\rho_{\ell}$. Obtain an expression for the depth, $h^{\prime}$ of the bottom surface of the container in terms of $P_{0}, \rho_{\ell}, g$ and $h$.

Figure 3 (a)

Figure 3 (b)

c) [5] A rod of length $\lambda$ and material density $\rho_{r}$ is pivoted by a hinge at a point on the wall of a container which has liquid of density $\rho_{\ell}$, below the level of the pivot point. The rod floats without touching the bottom of the container. If a fraction $\lambda / n$ of the rod is above the surface, determine $n$ in terms of $\rho_{r}$ and $\rho_{\ell}$.

Figure 4 (a)

Figure 4 (b)

d) A solid circular cone of height $h$, mass $m$, radius of base, $r$, is floating in a liquid of density $\rho_{\ell}$ with its axis vertical and vertex downwards, as shown in Figure $\mathbf{5}$. In the liquid, a height $h / n$ is submerged below the surface of the liquid.

(i) Obtain an expression relating the mass of the cone to the fraction submerged, $\frac{h}{n}$, in terms of $r, h, n$ and $\rho_{\ell}$

The volume of a circular cone is given by $V=\frac{1}{3} \pi r^{2} h$.

(ii) A weight equal to the weight of the cone is placed on the base whilst it is floating, and it sinks until the base is just level with the liquid, and then it rises again. For this condition, show that

$$ n^{3}+n^{2}+n=7 $$

You may neglect any effects of the inertia and viscosity of the liquid. Energy considerations may be helpful.

Figure 5

a) [3] The dam shown in Fig. $\mathbf{1}$ is used to generate hydroelectric power. The difference in height between the water levels of the dam is $61 \mathrm{~m}$, and the peak flow rate of the water through the dam is $3380 \mathrm{~m}^{3} \mathrm{~s}^{-1}$. Estimate the maximum power output of the hydroelectric facility.

Figure 1: HEP dam.

credit: https://upload.wikimedia.org/wikipedia/commons/b/b2/Epa-archives_the_dalles_dam-cropped.jpg

b) [3] In practice not all of the gravitational potential energy of the water behind the dam is transferred by the generators, and the water emerges from the dam with a velocity of $8.0 \mathrm{~m} \mathrm{~s}^{-1}$. Calculate the efficiency of the energy transfer process from the water.

c) [10] The dam can be considered to be a concrete prism of triangular cross-section, with height $h=60 \mathrm{~m}$, a base of thickness $t=60 \mathrm{~m}$, and a width of $w=2700 \mathrm{~m}$. The cross-section of the dam is illustrated in Figure 2. Points $\mathbf{F}$ and $\mathbf{C}$ are at the bottom of the dam. The water level is at the top of the dam wall.

Figure 2

(i) Calculate the resultant force on the face of the dam due to water pressure in terms of $h, w, \rho$ and $g$, where $\rho$ is the density of water.

(ii) Calculate the resultant moment (torque) about $\mathbf{F}$ due to this force, and calculate the effective height at which the resultant force acts to produce this moment. The centre of mass of a uniform triangle is at the point of intersection of its medians (the lines joining its vertices to the midpoints of the opposite sides). Thus the centre of mass lies two-thirds of the way along each median.

(iii) For the dam, state the position of the centre of mass of the dam relative to $\mathbf{F}$.

(iv) Calculate the two turning moments acting on the dam about $\mathbf{C}$, due to the water pressure and the weight of the dam.

(v) The frictional force of the dam on the ground beneath is given by $f=\mu N$ where $N$ is the normal reaction force of the ground, and $\mu=0.75$. Calculate the minimum density of concrete that could be used so that the dam will not slide over the ground.

(vi) Calculate the minimum density of concrete used to construct the dam so that it will not tip up about point $\mathbf{C}$.

Density of water is $\rho_{w}=1000 \mathrm{~kg} \mathrm{~m}^{-3}$

d) There is a significant danger of cracks forming in dams, leading to catastrophic failure with little or no warning. One form of non-destructive testing uses ultrasonic waves to investigate the interior of the concrete.

When a wave strikes a boundary (interface) between two different media it has to obey two constraints.

• the displacement of the material by the wave must be the same either side of the boundary.
• the energy transferred into the boundary must be the same as the total energy reflected and transmitted.

If an incident wave has an amplitude $A_{i}$, the reflected wave amplitude $A_{r}$ and the transmitted wave amplitude $A_{t}$, then we can write:

$$ A_{i}+A_{r}=A_{t} \quad (1) $$

The energy of a wave is proportional to the square of the amplitude, $A^{2}$, and the rate of energy flow in a medium is given by $\frac{\Delta E}{\Delta t}=\omega^{2} Z A^{2}$, where $Z$ stands for the acoustic impedance and is a parameter of the medium, and $\omega$ is a constant (it cancels in all our equations and can be ignored).

(i) From the energy flow through a boundary from medium 1 to medium 2 , show that

$$ Z_{1} A_{i}^{2}-Z_{1} A_{r}^{2}=Z_{2} A_{t}^{2} \quad (2) $$

where $Z_{1}$ and $Z_{2}$ are the impedances of the media either side of the boundary, as illustrated in Figure 3 .

Using these two requirements (eq. (1) and (2)) for a wave crossing an interface between two media at normal incidence:

(ii) show that $\frac{A_{r}}{A_{i}}=\frac{\left(Z_{1}-Z_{2}\right)}{\left(Z_{1}+Z_{2}\right)}$, and

(iii) derive an expression for $\frac{A_{t}}{A_{i}}$ in terms of $Z_{1}$ and $Z_{2}$.

(iv) If $Z_{1}=Z_{2}$, what can be said about the transmitted and reflected waves and the values of $\frac{A_{r}}{A_{i}}$ and $\frac{A_{t}}{A_{i}}$ ?

Figure 3

(v) When $Z_{2}>Z_{1}$, what effect does this produce?

The energy reflected back can be used to charaterise cracks in the concrete. The coefficient of reflection, $R$, is the ratio of the reflected to the incident energy flow, such that $R=\frac{A_{r}^{2}}{A_{i}^{2}}$. The coefficient of transmission, $T$, is the ratio of the transmitted energy flow in medium 2 to the incident energy flow in medium $1 .$

(v) Obtain expressions for $R$ and $T$ in terms of $Z_{1}$ and $Z_{2}$.

(vi) Calculate a value for $R+T$.

For a medium, $Z=\rho v_{\text {sound }}$, where $\rho$ is the density of the medium, and $v_{\text {sound }}$ is the speed of sound in the medium. Values for the densities and sound speeds in air, concrete and water are given in Table 1 .

(vii) Calculate the percentage of ultrasound energy reflected from a crack in the concrete when

i. the crack is filled with air,

ii. the crack is filled with water.