Potential Energy of Gravity

The Potential Energy of Gravity calculator computes the potential energy in a two mass (m₁ and m₂) system separated by a specified distance (r) from the center of masses. It is typically used when calculating the Potential Energy of Gravity due to celestial bodies.

INSTRUCTIONS: Choose units and enter the following:

• (m₁) This is the mass of object 1
• (m₂) This is the mass of object 2
• (r) This is the distance between the objects.

Potential Energy of Gravity (U_G): The energy is returned in joules.   However, this can be automatically converted to compatible units via the pull-down menu.

The Math / Science:

The Potential Energy of Gravity is originally derived from the equation for the Force of Gravity:

• `F=((G*M*m)/R^2)`

where,

• (F) is the Force
• (G) is the gravitational constant
• (M) is the mass of object 1 (typically the mass of the earth)
• (m) is the mass of object 2
• (R) is the separation of the two objects

Then, physics tells us that work is the integral of force. So by the relation of `W=intFdx` we can get a work value of:

`W= -((G*M*m)/(r_2-r_1))`

where,

• (F) is the Force
• (G) is the gravitational constant
• (M) is the mass of object 1 (typically the mass of the earth)
• (m) is the mass of object 2
• (r₂-r₁) is the separation of the two objects

Lastly, if we specifically look at the Potential Energy of gravity and we use the knowledge that `PE=-W` then we can substitute our equation for Work for Potential Energy. But if we think a little more we realize that the original force of gravity equation is used for gravity anywhere in the solar system. Since we want to know the force of gravity relative to our earth, then `r_1` is essentially at a length of infinity. Since `r_1` is under a fraction, then `r_1 = 0` so we are left with:

`PE = -((G*M*m)/r)`

where

• (PE) is the Potential Energy (sometimes denoted as `U`)
• (G) is the gravitational constant
• (M) is the mass of object 1 (typically the mass of the earth)
• (m) is the mass of object 2
• (r) is the distance of object 2 from the earth

Note: vCalc provides special mass and distance units for application in space sciences.  The mass units include Earth Mass, Jupiter Mass and Solar Mass.  The distances include Astronomical Units (ua), Light Seconds, Minutes, Hours, Days and Years, Parsecs, Kilo-Parsecs and Kilo-Light Years. 

Energy Calculators

• Kinetic Energy:                                    `KE= 1/2 *m * v^2`
• Kinetic Energy (change of velocity):    `KE = 1/2*m*(V_1-V_2)^2`
• Relativistic Kinetic Energy:                    `E_K = (m*c^2)/sqrt(1 - v^2"/"c^2") - m*c^2`
• Potential Energy:                                  `U(y) = m*g*y`
• Potential Energy of Gravity (two bodies):   `U(G) = - (G*m1*m2)/r`
• Nuclear Binding Energy:                       `E = m*c^2`
• Quantum Energy (Planck's Equation):   `E = h*f`
• Energy of a Particle in a Box:                `E_n=(n^2h^2)/(8mL^2)`
• Molecular Kinetic Energy:                      `KE = 3/2 * k_B *T`
• Electrostatic Potential Energy:               `E_(el) = k_e * (Q_1*Q_2)/d`
• Photon Energy from Wavelength:          `E = (h*c)/lambda`
• Heat Energy to Change Material Temperature:  `Q = C * m *DeltaT`