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10/07/09 - Potential Energy and Power
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 | Energy can be stored in an elastic band by stretching it or in a spring by stretching or compressing it.
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| We find that the force it takes to stretch an elastic band or to stretch (or compress) a spring increases the more we stretch (or compress) it.
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| The typical relationship gives us a Force-Stretch graph such as that shown below (known as Hooke's Law). Note that the slope of the straight line is given by k, and k is a measure of how "stiff" the spring is (or how hard it is to stretch). If it doesn't take much force to stretch a spring, then k is a small number (in Newtons/meter). If it takes a lot of force to stretch a spring, then k is a large number.
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| In a case like this, where the force is changing as we stretch the spring, we calculate the work done by finding the area under the Force-Stretch graph.
(i.e., multiplying the average force times the displacement). In this case, the average force is the average of F = 0 and F = kx, in other words, Favg = (0 + kx)/2 = (1/2)kx. The displacement is x, so the work done in stretching the spring is W = (1/2)(kx)(x) = (1/2)kx2.
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 | Gravitational Potential Energy
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| Gravitational attraction between two masses can be thought of as stretching an "invisible spring", except the force is described by Newton's Law of Gravitation:
The negative (-) sign here indicates that the gravitational force is attractive. A graph of the gravitational force F versus separation r is shown below:
Again, the area between the Force-separation curve and the r-axis is the potential energy. In this case it takes calculus to find the area, but the result is that two masses separated by a distance r have potential energy given by:
 .
If we're near the Earth's surface, we can just use PE = mgh, but if we're far enough away from the Earth's center for the gravitational force to weaken appreciably, we need to use the formula given above. Note that this formula agrees with Kepler's law stating that planet's orbiting the sun slow down as they move away from the sun and speed up as they approach the sun. As they move away, some of their kinetic energy is converted to potential energy and they slow down. As they approach the sun, some of their potential energy is converted back to kinetic energy and they speed up.
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| The total energy of a system consists of the sum of potential energy of various forms and kinetic energy of various forms.
Total Energy = Potential Energy + Kinetic Energy We find that total energy is constant unless work is done by the system or on the system.
Work done on a system increases its total energy, and work done by a system decreases its total energy.
If no work is done on a system, the total energy is unchanged, which means that an increase in potential results in a decrease in kinetic energy by the same amount, and a decrease in potential energy results in an increase in kinetic energy by the same amount.
∆Etotal = 0 means ∆PE + ∆KE = 0 or ∆KE = -∆PE. Another way to say that the total energy is constant is:
PE1 + KE1 = PE2 + KE2 where PE1 and KE1 are the initial potential and kinetic energies of a system, and PE2 and KE2 are the potential and kinetic energies of a system at a later time.
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| Example of Conservation of Energy using Gravitational Potential Energy: Earth's orbit around the Sun
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| The Earth's orbit is quite circular. Using the radius of the Earth's orbit (R = 1.5E11 m) and the time it takes for one revolution around the Sun (T = 1 year = 3.15E7 seconds), we can calculate the speed of the earth in its orbits as v = (2πR/T) = 29865 m/s. We can then calculate the total energy of the Earth's orbit by combining its kinetic energy and its potential energy:
If we increased the Earth's velocity by 20% when it is near the Sun (to 35838.4 m/s), then the total energy would be:
From Kepler's Second Law, the speed of a planet at its nearest and farthest distances from the Sun are related to the speeds of the planets at those points by the relationship:
Since the total energy is constant, we can say:
Substituting the values in for the various quantities we know and doing lots of algebra, we find that:
and
With this information we can graph the Earth's orbit using its current values of R and v (for the circular orbit shown below) and the elliptical orbit it would have if its orbital velocity increased by 20% when it was near the Sun:
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| Power is a measure of how fast work is done or energy is expended: P = ∆E/∆t.
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| The SI unit of power is the Joule/second or Watt.
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| A 100-W light bulb uses 100 Joules of energy every second.
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| 1 horsepower = 1hp = 746 W.
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| Read pp. 182-197 of the text.
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