Let p be a point at a distance d from the center of a circle of radius r. the curve traced out by p as the circle rolls along a straight line is called a trochoid. (think of the motion of a point on a spoke of a bicycle wheel.) the cycloid is the special case of a trochoid with d = r. using the same parameter θ as for the cycloid and, assuming the line is the x-axis and θ = 0 when p is at one of its lowest points, parametric equations of the trochoid are x = rθ − d sin θ y = r − d cos θ. find the area under one arch of the trochoid found above for the case d < r.

Contrast the force of gravity between these pairs of objects a 1 kg mass and a 2 kg mass that are 1 m apart and two 2 kg masses that are 1 m apart

Aheat engine running backward is called a refrigerator if its purpose is to extract heat from a cold reservoir. the same engine running backward is called a heat pump if its purpose is to exhaust warm air into the hot reservoir. heat pumps are widely used for home heating. you can think of a heat pump as a refrigerator that is cooling the already cold outdoors and, with its exhaust heat qh, warming the indoors. perhaps this seems a little silly, but consider the following. electricity can be directly used to heat a home by passing an electric current through a heating coil. this is a direct, 100% conversion of work to heat. that is, 19.0 \rm kw of electric power (generated by doing work at the rate 19.0 kj/s at the power plant) produces heat energy inside the home at a rate of 19.0 kj/s. suppose that the neighbor's home has a heat pump with a coefficient of performance of 4.00, a realistic value. note: with a refrigerator, "what you get" is heat removed. but with a heat pump, "what you get" is heat delivered. so the coefficient of performance of a heat pump is k=qh/win. an average price for electricity is about 40 mj per dollar. a furnace or heat pump will run typically 200 hours per month during the winter. what does one month's heating cost in the home with a 16.0 kw electric heater? what does one month's heating cost in the home of a neighbor who uses a heat pump to provide the same amount of heating?

He latent heat of vaporization for ethyl alcohol is 854 j/g. the amount of energy, rounded to the nearest whole number, needed to change 5.20 grams of ethyl alcohol from a liquid to a gas is