Chapter 11

If a 70 kg person climbs 10 flights of stairs, each flight $3\mathrm{m}$ high, how much potential energy have they gained?

If an $80\mathrm{kg}$ person is capable of delivering external mechanical energy at a rate of $200\mathrm{W}$ sustained over several minutes, ${}^{34}$ how high would they be able to climb in two minutes?

A $10\mathrm{kg}$ box is lifted $2\mathrm{m}$ off the floor and placed on a frictionless horizontal conveyor to take it $30\mathrm{m}$ across a warehouse. At the end of the conveyor, it is lowered $1\mathrm{m}$ where it ends up on a shelf. How much net gravitational potential energy was given to the box from the start to the end of the process?

A gallon of gasoline contains about $130\mathrm{MJ}$ of chemical energy at a mass of around $3\mathrm{kg}$. How high would you have to lift the gallon of gasoline to get the same amount of gravitational potential energy? Compare the result to the radius of the earth.

A typical American household uses approximately $30\mathrm{kWh}$ per day of electricity. Convert this to Joules and then imagine building a water tank $10.8\mathrm{m}$ above the house ${}^{37}$ to supply one day's worth of electricity. ${}^{38}$ How much mass of water is this, in kg? At a density of $1,000\mathrm{kg}/{\mathrm{m}}^3$, what is the volume in cubic meters, and what is the side length of a cube ${}^{39}$ having this volume? Take a moment to visualize (or sketch) this arrangement.

The biggest hydroelectric installation in the U.S. is the Grand Coulee dam on the Columbia River. The enormous flow rate reaches its maximum at $4,300{\mathrm{m}}^3/\mathrm{s}$, and the dam (reservoir) height is $168\mathrm{m}$. At an efficiency of $90\mathrm{\%}$, at what rate is this dam capable of producing hydroelectric power (in ${\mathrm{GW}}^{40}$ )? Don't forget the density of water and that $g\approx 10\mathrm{m}/{\mathrm{s}}^2$.

A dam 50 meters high is constructed on a river and is delivering $180\mathrm{MW}$ at some moment in time. What is the flow rate of water, in cubic meters per second, if the facility converts gravitational potential energy into electricity at $90\mathrm{\%}$ efficiency?

A hydroelectric facility is built to deliver a peak power of $1\mathrm{GW}$, which it manages to do for three months of the year during the spring snow- melt. But for three months in summer, it drops to $700\mathrm{MW}$, then $500\mathrm{MW}$ for three months in fall. In winter, it drops way down to $200\mathrm{MW}$ for three months. Using the concept of the capacity factor (Definition 11.2.1), what is the annual average capacity factor for this facility?

While the Chief Joseph Dam on the Columbia River can generate as much as $2.62\mathrm{GW}(2.62\times {10}^9\mathrm{W})$ of power at full flow, the river does not always run at full flow. The average annual production is 10.7 TWh per year $(10.7\times {10}^{12}\mathrm{Wh}/\mathrm{yr})$. What is the capacity factor of the dam: the amount delivered vs. the amount if running at $100\mathrm{\%}$ capacity the whole year?

The Robert Moses Niagara dam in New York is rated at $2,429{\mathrm{MW}}^{41}$ and has a high capacity factor of $0.633.$ How many $\mathrm{kWh}$ does it produce in an average day, and how many homes would this serve at the national average of $30\mathrm{kWh}/$ day?