The rigid gate, $O A B$, of the given figure is hinged at $O$ and rests against a rigid support at $B$. What minimum horizontal force, $P$, is required to hold the gate closed if its width is $3 \mathrm{~m}$? Neglect the weight of the gate and friction in the hinge. The back of the gate is exposed to the atmosphere.

Find the magnitude and location of the net horizontal force on the gate shown. The gate width is $5.0 \mathrm{~m}$.

A rectangular gate is hinged at the water line, as shown in the given figure. The gate is $4\ \mathrm{ft}$ high and $10\ \mathrm{ft}$ wide. The specific weight of water is $62.4\ \mathrm{lbf} / \mathrm{ft}^3$. Find the necessary force (in $\mathrm{lbf}$ ) applied at the bottom of the gate to keep it closed.

Consider the general form of the Reynolds transport theorem (RTT) given by $rac{d B_{sys}}{d t}=rac{d}{d t} \int_{\mathrm{CV}} ho b d V+\int_{\mathrm{CS}} ho b ec{V}_{r} \cdot ec{n} d A$ where is the velocity of the fluid relative to the control surface. Let $B_{sys}$ be the mass m of a closed system of fluid particles. We know that for a system, dm/dt = 0 since no mass can enter or leave the system by definition. Use the given equation to derive the equation of conservation of mass for a control volume.

A rectangular gate having a width of $\mathrm{5} \mathrm{ft}$ is located on the sloping side of a tank as shown in the given figure. The gate is hinged along its top edge and is held in position by the force $P$. Friction at the hinge and the weight of the gate can be neglected. Determine the required value of $P$.

A gate, in the shape of a quarter-cylinder, hinged at $A$ and sealed at $B$, is $3 \mathrm{~m}$ wide. The bottom of the gate is $4.5 \mathrm{~m}$ below the water surface. Determine the force on the stop at $B$ if the gate is made of concrete; $R=3 \mathrm{~m}$.

The gate $A O C$ shown is $6\ \mathrm{ft}$ wide and is hinged along $O$. Neglecting the weight of the gate, determine the force in bar $A B$. The gate is sealed at $C$.

The hinged gate ABC closes an opening of width $b$ (perpendicular to the paper) in a water channel. The water has free access to the underside as well as the right side of the gate. When the water level rises above a certain value of $h$, the gate will open. Determine the critical value of $h$. Neglect the mass of the gate.

A rectangular gate (width $w=2 \mathrm{~m}$ ) is hinged as shown, with a stop on the lower edge. At what depth $H$ will the gate tip?

In Fig. P2.55, gate $AB$ is $5 \ \text{ft}$ wide into the paper, and stop $B$ will break if the water force on it equals $9200 \ \text{lbf}$. At what depth $h$, this condition will reach?

For a corner reflector to be effective, its surfaces must be precisely perpendicular. Suppose the surfaces of a corner reflector left on the Moon's surface by the Apollo astronauts formed a $90.001^{\circ}$ angle with each other. If a laser beam is bounced back to Earth from this reflector, how far (in kilometers) from its starting point will the reflected beam strike Earth? For simplicity, assume the beam reflects from only two sides of the reflector, and that it strikes the first surface at precisely $45^{\circ}$.

The flow of water from a reservoir is controlled by a $5$ - $\mathrm{ft}$-wide $\mathrm{L}$-shaped gate hinged at point $A$, as shown in the figure. If it is desired that the gate open when the water height is $12\ \mathrm{ft}$, find the mass of the required weight $W$.

A $0.5 \times 0.8-\mathrm{m}$ gate $A B$ is located at the bottom of a tank filled with water. The gate is hinged along its top edge $A$ and rests on a frictionless stop at $B$. Determine the minimum tension required in cable $B C D$ to open the gate.

The velocity field for a line vortex in the $r \theta$-plane is given by

$$u_r=0 \quad u_\theta=\frac{K}{r}$$

where $K$ is the line vortex strength. For the case with $K=1.5 \mathrm{~m} / \mathrm{s}^2$, plot a contour plot of velocity magnitude (speed). Specifically, draw curves of constant speed $V=0.5,1.0,1.5$, $2.0$, and $2.5 \mathrm{~m} / \mathrm{s}$. Be sure to label these speeds on your plot.