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Converting between potential energy and kinetic energy

Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to mgh size 12{ ital "mgh"} {} on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of PE g size 12{"PE" rSub { size 8{g} } } {} to KE size 12{"KE"} {} without explicitly considering the intermediate step of work. (See [link] .) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces.

(a) The weight attached to the cuckoo clock is raised by a height h shown by a displacement vector d pointing upward. The weight is attached to a winding chain labeled with a force F vector pointing downward. Vector d is also shown in the same direction as force F. E in is equal to W and W is equal to m g h. (b) The weight attached to the cuckoo clock moves downward. E out is equal to m g h.
(a) The work done to lift the weight is stored in the mass-Earth system as gravitational potential energy. (b) As the weight moves downward, this gravitational potential energy is transferred to the cuckoo clock.

More precisely, we define the change in gravitational potential energy Δ PE g size 12{Δ"PE" rSub { size 8{g} } } {} to be

Δ PE g = mgh , size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {}

where, for simplicity, we denote the change in height by h size 12{h} {} rather than the usual Δ h size 12{Δh} {} . Note that h size 12{h} {} is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is

mgh = 0.500 kg 9.80 m/s 2 1.00 m = 4.90 kg m 2 /s 2 = 4.90 J.

Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work .

Using potential energy to simplify calculations

The equation Δ PE g = mgh size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {} applies for any path that has a change in height of h size 12{h} {} , not just when the mass is lifted straight up. (See [link] .) It is much easier to calculate mgh size 12{ ital "mgh"} {} (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position h size 12{h} {} of a mass m size 12{m} {} is accompanied by a change in gravitational potential energy mgh size 12{ ital "mgh"} {} , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.

There is a four-story building. A person is carrying a television up the stairs of the building. The height of third story is h from the ground. A girl is standing outside the building and is lifting a similar television with the help of a pulley.
The change in gravitational potential energy ( Δ PE g ) size 12{ \( Δ"PE" rSub { size 8{g} } \) } {} between points A and B is independent of the path. Δ PE g = mgh size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {} for any path between the two points. Gravity is one of a small class of forces where the work done by or against the force depends only on the starting and ending points, not on the path between them.

The force to stop falling

A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly (with his knee joints compressing by 0.500 cm), calculate the force on the knee joints.

Strategy

This person’s energy is brought to zero in this situation by the work done on him by the floor as he stops. The initial PE g size 12{"PE" rSub { size 8{g} } } {} is transformed into KE size 12{"KE"} {} as he falls. The work done by the floor reduces this kinetic energy to zero.

Solution

The work done on the person by the floor as he stops is given by

W = Fd cos θ = Fd , size 12{W= ital "Fd""cos"θ= - ital "Fd"} {}

with a minus sign because the displacement while stopping and the force from floor are in opposite directions ( cos θ = cos 180º = 1 ) size 12{ \( "cos"θ="cos""180""°=" - 1 \) } {} . The floor removes energy from the system, so it does negative work.

The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through height h size 12{h} {} :

KE = Δ PE g = mgh , size 12{"KE"= - Δ"PE" rSub { size 8{g} } = - ital "mgh"} {}

The distance d size 12{d} {} that the person’s knees bend is much smaller than the height h size 12{h} {} of the fall, so the additional change in gravitational potential energy during the knee bend is ignored.

The work W size 12{W} {} done by the floor on the person stops the person and brings the person’s kinetic energy to zero:

W = KE = mgh . size 12{W= - "KE"= ital "mgh"} {}

Combining this equation with the expression for W size 12{W} {} gives

Fd = mgh . size 12{ - ital "Fd"= ital "mgh"} {}

Recalling that h size 12{h} {} is negative because the person fell down , the force on the knee joints is given by

F = mgh d = 60.0 kg 9.80 m /s 2 3 . 00 m 5 . 00 × 10 3 m = 3 . 53 × 10 5 N. size 12{F= - { { ital "mgh"} over {d} } = - { { left ("60" "." 0" kg" right ) left (9 "." "80"" m/s" rSup { size 8{2} } right ) left ( - 3 "." "00"`m right )} over {5 "." "00" times "10" rSup { size 8{ - 3} } " m"} } =3 "." "53" times "10" rSup { size 8{5} } `N "." } {}

Discussion

Such a large force (500 times more than the person’s weight) over the short impact time is enough to break bones. A much better way to cushion the shock is by bending the legs or rolling on the ground, increasing the time over which the force acts. A bending motion of 0.5 m this way yields a force 100 times smaller than in the example. A kangaroo's hopping shows this method in action. The kangaroo is the only large animal to use hopping for locomotion, but the shock in hopping is cushioned by the bending of its hind legs in each jump.(See [link] .)

Questions & Answers

Application of nanotechnology in medicine
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Source:  OpenStax, Work and energy. OpenStax CNX. Nov 09, 2015 Download for free at http://legacy.cnx.org/content/col11902/1.1
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