Graviational Potential Energy
CLAIMED BY NICK RICCARDELLI, Cooper Link (Fall 2017)
The Main Idea
Gravitational Potential Energy is a measure of the energy stored in an object due to its interaction with a larger object. The most common example of Gravitational Potential Energy is the interaction between an object close to the surface of the earth and the earth. In this situation, the gravitational acceleration constant can be assumed to be about 9.8 m/s^2. The equation for GPE is given as
PE_Grav (J) = Weight (kg) * Constant (9.8 m/s^2) * Height (M)
Change in Gravitational Potential Energy (ΔPE) is a representation of the change in height, or distance from the system's chosen zero point. This is because both the mass of a given object and the gravitational constant do not change in problems of this nature.
The expression for gravitational potential energy comes from the Law of Gravity, and therefore is equal to the amount of work done against gravity to change a given object's distance from the zero point of its system. The form of gravitational potential energy used when not operating on Earth's Surface is the expression that follows:
U = -GMm/r
In this expression, GPE is given as U, G is the gravitational constant outside of the Earth's sphere of influence (6.754*10^-11 m^3 kg^-1 s^-2). Both M's stand for the mass of an object and although they are interchangeable, general convention says big M stands for mass of the attracting object and little m stands for mass of the object attracted. Finally, r is given as the distance between the center of gravity of the two objects.
A Mathematical Model
PE = mgh
- m = the object's mass in kilograms
- g = gravitational acceleration, 9.8 m/s^2 on earth
- h = the object's height in meters from a chosen reference point
U = -GMm/r
- U = GPE
- M = Mass of Attracting Object
- m = Mass of Object Attracted
- r = Distance between two center of masses
- G = Gravitational Acceleration (6.754*10^-11 m^3 kg^-1 s^-2)
Examples
What is the potential energy of a 5kg object on top of a 20 meter structure?
- PE = mgh
- PE = (5 kg) * (20 m) * (9.8 m/s^2)
- PE = 980 kg*m^2/s^2 = 980 J
A group of skydivers are going up for a jump. With all the gear on, one diver weighs 100 kg. Starting from the ground, their plane takes them up to a height of 6000 m. What is the change in potential energy in this system?
PE = mg(Δh) m = 100 kg g = 9.81 m/s^2 h = 6000 - 0 m
Therefore, PE = 100 kg * 9.81 m/s^2 * 6000 m which means that ΔPE is equal to 5886000 J.
This presents us with an interesting issue, even when calculating a simple ΔPE we are given a gigantic change in energy, or Joules. To solve this we use a different measure of energy, Kilo-joules. This simply cleans up our solution and makes it easier to work with in the future. Using simple dimensional analysis we see that our new answer is 5886 kJ.
One thing that GPE outside of the Earth's sphere of influence is particularly useful for is calculating escape velocity. Now, this utilizes GPE in what is known as a work equation. Stemming from conservation of energy, this equation simply gives us a way to show energy being converted from potential to kinetic.
The equation for escape velocity is given as follows
KE = PE
1/2 m V^2 - GMm/r = 0 First step, bring GPE over. 1/2 m V^2 = GMm/r next, we can cancel little m from the equation. Stop now and consider what this means when calculating escape velocity....
1/2 v^2 = GM/r
That's right! This means that regardless of the mass of an object, it will have the same escape velocity. Whether you're throwing a baseball or launching a rocket, escape velocity is constant!
Finally, bring over the 1/2 and take the square root of the entire thing to find that. Escape Velocity = sqrt(2GM/r)
And since speed is the magnitude of velocity, we can tell that we should take the positive square root to determine the speed at which an object must travel in order to leave the sphere of a bodies influence.
Connectedness
Gravitational Potential Energy deals directly with one of the properties of matter, mass. Because of this, it is near omnipresent in physics. When calculating the work done on a system, or the potential increase of a system, GPE is almost always used. One important thing to remember is that GPE is used in two different ways; on the Earth's surface GPE = mgh where g is 9.8m/s^2. However, when we are considering a system off the surface of the Earth we are forced to use something a little more complicated in this law for GPE - U = -GMm/r
Because of conservation of momentum, gravitational potential energy has many applications in physics, chemsistry, and several facets of engineering. For example, how fast will a ball hit the ground when dropped from a certain height? How much work is done by a tall waterfall? How high can you build a rollercoaster? Will roadrunner escape the perils of introductory physics (see video below)?
Video
References
- Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4nd Edition by R. Chabay & B. Sherwood (John Wiley & Sons 2015)
- Hyperphysics: Gravitational Potential Energy, Georgia Southern University