Gravitational Potential Energy


Gravitational potential energy is the energy an object has because of its mass and its position in a gravitational field.

The gravitational potential energy of an object is:

  • directly proportional to the mass of the object
  • directly proportional to the height of the object above a reference point
  • directly proportional to the value of the gravitational field it is in.

The equation we use to find the gravitational potential energy of an object is:

U=mgh

Where:

U= potential energy in joules (J)

m= mass in kilograms (kg)

g= gravitational field in \cfrac { m }{ s^{ 2 } } . On Earth, the average value of this field at sea level is 9.8\cfrac { m }{ s^{ 2 } }

h= height of the object above the reference point in metres (m). The reference point is often the ground or another surface off which an object is lifted.

Not all objects will be lifted off the ground. For example, a problem may consider the gravitational potential energy gained if a crane lifts an object from 10m to 25m. To find the change in gravitational potential energy of an object when it moves from one altitude to another (close to the Earth’s surface) we use:

U=mg\Delta h

The equations above only apply for objects that are close to the Earth’s surface. This is because the value of the gravitational field reduces at a significant distance above the Earth’s surface.


Example 1:

A 5kg object is lifted from the ground to a height of 2.3m. Calculate the change in gravitational potential energy of the object:

As the object is lifted from the ground:

U=mgh

U=5\times 9.8\times 2.3

U=112.7\quad J


Example 2:

A 200kg object is lifted from 10m to 25m. Calculate the change in gravitational potential energy of the object:

As the object is lifted from one altitude to another:

U=mg\Delta h

U=200\times 9.8(25-10)

U=29400\quad J

U=29.4\quad kJ


Example 3:

What height must a 1000kg wrecking ball be lifted to if it is to gain 68.6 kJ of energy:

The object has to gain 68.6 kJ of energy, which = 68600 J of energy.

As the object is lifted from the ground:

U=mgh

68600=1000\times 9.8\times h

h=\cfrac { 68600 }{ 1000\times 9.8 }

h=7\quad m

 

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