The 60-W fan of a central heating system is to circulate air through the ducts. The analysis of the flow shows that the fan needs to raise the pressure of air by 50 Pa to maintain flow. The fan is located in a horizontal flow section whose diameter is 30 cm at both the inlet and the outlet. Determine the highest possible average flow velocity in the duct.

Find $\frac{d y}{d x}$. Assume $\frac{d x}{d t} \neq 0$. $x(t)=\cos ^{3} t, \quad y(t)=\sin ^{3} t$

What is the area moment of inertia of a circle?

Consider a fan located in a 3 ft $\times 3$ ft square duct. Velocities at various points at the outlet are measured, and the average flow velocity is determined to be 22 ft/s. Taking the air density to $0.075 \mathrm{lbm} / \mathrm{ft}^{3}$, estimate the minimum electric power consumption of the fan motor.

Water is pumped from a lake to a storage tank $20 \mathrm{~m}$ above at a rate of $70 \mathrm{~L} / \mathrm{s}$ while consuming $20.4 \mathrm{~kW}$ of electric power. Disregarding any frictional losses in the pipes and any changes in kinetic energy. Find (a) the overall efficiency of the pump-motor unit and (b) the pressure difference between the inlet and the exit of the pump.

A water pump that consumes 2 kW of electric power when operating is claimed to take in water from a lake and pump it to a pool whose free surface is 30 m above the free surface of the lake at a rate of 50 L/s. Determine if this claim is reasonable.

The driving force for fluid flow is the pressure difference, and a pump operates by raising the pressure of a fluid (by converting the mechanical shaft work to flow energy). A gasoline pump is measured to consume 3.8 kW of electric power when operating. If the pressure differential between the outlet and inlet of the pump is measured to be 7 kPa and the changes in velocity and elevation are negligible, determine the maximum possible volume flow rate of gasoline.

Determine the power required for a $1150$-$~\mathrm{kg}$ car to climb a $100$-$~\mathrm{m}$-long uphill road with a slope of $30^\circ$ (from horizontal) in $12 \mathrm{~s}$

(a) at a constant velocity,

(b) from rest to a final velocity of 30 m/s, and

(c) from 35 m/s to a final velocity of 5 m/s. Disregard friction, air drag, and rolling resistance.

Consider a room that is initially at the outdoor temperature of 20$^\circ{}$C. The room contains a 40-W lightbulb, a 110-W TV set, a 300-W refrigerator, and a 1200-W iron. Assuming no heat transfer through the walls, determine the rate of increase of the energy content of the room when all of these electric devices are on.

Consider a 1400-kg car cruising at constant speed of 70 km/s. Now the car starts to pass another car, by accelerating to 110 km/h in 5 s. Determine the additional power needed to achieve this acceleration. What would your answer be if the total mass of the car were only 700 kg?

A university campus has 200 classrooms and 400 faculty offices. The classrooms are equipped with 12 fluorescent tubes, each consuming 110 W. including the electricity used by the ballasts. The faculty offices, on average, have half as many tubes. The campus is open 240 days a year. The classrooms and faculty offices are not occupied an average 4 h a day, but the lights are kept on. If the unit cost of electricity is $0.11/kWh, determine how much the campus will save a year if the lights in the classrooms and faculty offices are turned off during unoccupied periods.

The basic barometer can be used as an altitude-measuring device in airplanes. The ground control reports a barometric reading of 753 mmHg while the pilot’s reading is 690 mmHg. Estimate the altitude of the plane from ground level if the average air density is $1.20 \mathrm{kg} / \mathrm{m}^{3}$.

At $45^\circ$ latitude, the gravitational acceleration as a function of elevation z above sea level is given by g = a ‒ bz, where $a = 9.807 m/s^2$ and $b = 3.32 imes 10^{‒6} s^{‒2}.$ Determine the height above sea level where the weight of an object will decrease by 1 percent.

A river flowing steadily at a rate of $175 \mathrm{m}^{3} / \mathrm{s}$ is considered for hydroelectric power generation. It is determined that a dam can be built to collect water and release it from an elevation difference of 80 m to generate power. Determine how much power can be generated from this river water after the dam is filled.