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## Work

Work transfers energy to a system through the forceful manipulation of its state. Do some work on yourself to master this fundamental mode of energy transfer.

# Level 2

Water in a river speeds up as it goes over the top of a waterfall. What is the primary reason this happens?

Consider a village of area 0.16 square kilometers on the equator. The total energy intensity of sunlight at the surface of the earth when the sun is directly overhead is $$1370\text{ W/m}^2$$. How much solar energy in Joules hits the village per second at noon?

Bicyclists and other things that go fast must overcome air resistance, or drag, even to maintain a constant speed. A simple empirical model for the drag force on an object when the air flows smoothly around the object is $$\vec{F}_{Drag}=-c\vec{v}$$, where $$\vec{v}$$ is the velocity of the object and $$c$$ is a constant that depends on the size and shape of the object. Consider a bicyclist putting out some power $$P_0$$ (in watts) to overcome drag and maintain some constant speed $$v_0$$. She then increases her speed to $$1.2 v_0$$, which requires her to put out a power $$P_1$$ to maintain. What is the ratio $$\frac{P_1}{P_0}$$?

Suppose that to hike a distance $$L$$ while carrying nothing, your body requires an amount $$L \gamma_0$$ of food, and that when you carry an extra weight $$M$$ in your backpack, your body is less efficient and requires the amount of food $L \gamma_0 \left(1+\frac{M}{W_\text{body}}\right)$ where $$W_\text{body}$$ is the weight of your body.

If you start out with the weight $$M$$ of food in your ultralight (weightless) backpack, how far (in miles) can you hike in total?

Assumptions

• All your energy comes from eating the food in your backpack.
• You eat the food as needed for your energy needs.
• $$M$$ = 50 lb
• $$W_\text{body}$$ = 160 lb
• $$\gamma_0=12^{-1}$$ lb food per mile

There is a chain of uniform density on a table with negligible friction. The length of the entire chain is $$\si{1\ \meter}$$. Initially, one-third of the chain is hanging over the edge of the table. How long will it take the chain (in seconds) to slide off the table?

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