In Fig. P2.71 gate $A B$ is $3 \mathrm{~m}$ wide into the paper and is connected by a rod and pulley to a concrete sphere $(\mathrm{SG}=$ 2.40). To keep the gate closed, what will be the sufficient diameter of the sphere ?

Suppose that wet concrete $\left(\gamma=150 \mathrm{lbf} / \mathrm{ft}^3\right)$ behaves as a liquid. Find the force per unit foot of length exerted on the forms. If the forms are held in place as shown, with ties between vertical braces spaced every $2 \mathrm{ft}$, what force is exerted on the bottom tie?

Gate $A B$ in Fig. P2.65 is semicircular, hinged at $B$, and held by a horizontal force $P$ at $A$. Determine the force $P$ is required for equilibrium.

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?

The tank in Fig. P2.90 is $120 \mathrm{~cm}$ long into the paper. Determine the horizontal and vertical hydrostatic forces on quarter-circle panel $AB$ . The fluid is water at $20 \degree C$. Neglecting atmospheric pressure.

Calculate the magnitude and direction of the resultant hydrostatic force the water exerts on the parabolic face $A B$ of the wall if it is $3 \mathrm{~m}$ wide.

A vertical rectangular gate is 8 ft wide and 10 ft long and weighs 6000 lb. The gate slides in vertical slots in the side of a reservoir containing water. The coefficient of friction between the slots and the gate is 0.03. Determine the minimum vertical force required to lift the gate when the water level is 4 ft above the top edge of the gate.

Find the planar density and packing fraction for FCC nickel in the $(100)$, $(110)$, and $(111)$ planes. Which, if any, of these planes are close packed?

The swing-check valve in Fig. P2.70 covers a $22.86$-$\mathrm{cm}$ diameter opening in the slanted wall. The hinge is $15 \mathrm{~cm}$ from the centerline, as shown. The valve will open when the hinge moment is $50 \mathrm{~N}$-$\mathrm{m}$. Determine the value of $h$ for the water to cause this condition.

A long, vertical wall separates seawater from freshwater. If the seawater stands at a depth of $7 \mathrm{~m}$, what depth of freshwater is required to give a zero resultant force on the wall? When the resultant force is zero will the momentum due to the fluid forces be zero? Explain.

A concrete form used to pour a basement wall is to hold wet concrete mix ( $\mathrm{sg}=2.6$ ) during construction. The wall is to be $3 \mathrm{~m}$ high, $10 \mathrm{~m}$ long, and $150 \mathrm{~mm}$ thick. What pressure does the wet concrete exert at the bottom of the form?

A horizontal 2-m-diameter conduit is half filled with a liquid (SG = 1.6) and is capped at both ends with plane vertical surfaces. The air pressure in the conduit above the liquid surface is 200 kPa. Determine the resultant force of the fluid acting on one of the end caps, and locate this force relative to the bottom of the conduit.

A circular 2-m-diameter gate is located on the sloping side of a swimming pool. The side of the pool is oriented $60^{\circ}$ relative to the horizontal bottom, and the center of the gate is located 3 m below the water surface. Determine the magnitude of the water force acting on the gate and the point through which it acts.

Consider a Carnot cycle executed in a closed system with air as the working fluid. The maximum pressure in the cycle is 1300 kPa while the maximum temperature is 950 K. If the entropy increase during the isothermal heat rejection process is $0.25 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}$ and the net work output is 100 kJ/kg, determine (a) the minimum pressure in the cycle, (b) the heat rejection from the cycle, and (c) the thermal efficiency of the cycle. (d) If an actual heat engine cycle operates between the same temperature limits and produces 5200 kW of power for an air flow rate of 95 kg/s, determine the second law efficiency of this cycle.