Potential Energy
Potential energy is the ability an object possesses to cause change based on its position. By winding a spring it has more ability to do something than when it is unwound. The spring is not actually doing anything, but it could. This is the nature of potential energy.
The potential energy we use the most is gravitational potential energy. This is the energy an object has due to its position from the center of the earth. If there were no gravitational force, there would be no gravitational potential energy. Since work causes a change in potential energy, the amount of gravitational potential energy an object has depends on the work done on it to change its vertical position. If you move an object sideways the grqavitational potential energy of the object does not change since it is the same distance from the center of the earth. 
Potential energy only changes when the height of an object changes. It does not matter the path the object takes to get there. Just look at the beginning and ending heights to calculate the change in energy. The formula for calculating the gravitational potential energy of an object is: PE = m g h
Where h is the height above or below the zero reference line. This reference line can be placed anywhere it is convenient in the problem. Usually, ground level is chosen as the reference line. 
The unit of energy was renamed to make it a shorter unit. The Kg m2/s2 is also a Joule (J) of energy. Notice that to get Joules of energy the mass must be in kilograms and the height in m. Since work causes an equal change in energy it makes sense that they would have the same unit of measurement. If you have nothing better to do you can prove to yourself that a Nm and a Kgm2/s2 are equivalent. EX. What is the gravitational potential energy of a 1.6 Kg book on a shelf 2.0 m above the floor? PE = m g h = 1.6 Kg (9.8 m/s2 )(2.0 m) = 31.4 J (relative to the floor)
EX. If a 1.5 Kg book is resting on a table 0.8 m above the floor, what is its potential energy
relative to a) the floor,
b) the table, and
c) the ceiling which is 1.8 m above?
a) in this case the floor would be the zero reference height.
PE = mgh = 1.5 Kg (9.8 m/s2 )(0.8 m) = 11.8 J b) in this case the table is the zero reference height. Since the table is at that height, h = 0
so PE = 0 J. c) in this case the ceiling is the zero reference height. Since the book is below the zero line,
its h is negative.
PE = mgh = 1.5 Kg (9,8 m/s2 )(-1.8 m) = -26.5 J
A change in potential energy can be calculated from the change in height the object undergoes.
DPE = PEf - PEo = mghf - mgho = mg (hf - ho)
EX. What is the change in potential energy of a 30 Kg child on a swing if their highest point of the swing is 2.4 m above the ground and the lowest point of the swing is 0.6 m off the ground? 
DPE = mg (hf - ho) = 30 Kg (9.8 m/s2 )(2.4 m - 0.6 m) = 529 J
This would be a gain in potential energy as they go up and a loss of potential energy as they
come down.
Although potential energy can have both positive and negative values, it is not a vector quantity. You have to keep track of the gains or losses in energy, but the usual math procedures can be used. No vector treatment is required.
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