Asmall mass m hangs at rest from a vertical rope of length ℓ that is xed to the ceiling. a force then pushes on the
mass, perpendicular to the taut rope at all times, until the rope is oriented at an angle θ = θ0 and the mass has been
raised by a vertical distance h. assume the force’s magnitude f is adjusted so that the mass moves at constant
speed along its curved trajectory. show that the work done by during this process equals mgh, which is equivalent to the amount of work it takes to slowly lift a mass m straight up by a height h. [hint: when the angle is increased by dθ (in radians), the mass moves along an arc length ds = ℓ dθ.]

f is prop to m1xm2 double m1 double force

so here is the information for current research:

-nrel's geothermal analysts, scientists, and engineers perform a full range of research from developing advanced methods of locating and accessing geothermal resources to exploring innovative technologies to extract and produce clean energy from the heat of the earth

and here is the information for the technology of geothermal energy:

-it's clean and sustainable

-resources of geothermal energy range from the shallow ground to hot water and hot rock found a few miles beneath the earth's surface, and down even deeper to the extremely high temperatures of molten rock called magma

-almost everywhere, the shallow ground or upper 10 feet of the earth's surface maintains a nearly constant temperature between 50° and 60°f (10° and 16°c)

-geothermal heat pumps can tap into this resource to heat and cool buildings

-a geothermal heat pump system consists of a heat pump, an air delivery system (ductwork), and a heat exchanger-a system of pipes buried in the shallow ground near the building

-in the winter, the heat pump removes heat from the heat exchanger and pumps it into the indoor air delivery system

-in the summer, the process is reversed, and the heat pump moves heat from the indoor air into the heat exchanger

-the heat removed from the indoor air during the summer can also be used to provide a free source of hot water

grav potential energy

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