Gravitational Potential Energy: Gravitational potential energy is the energy acquired by an object due to a shift in its position when it is present in a gravitational field. In simple terms, it can be stated that gravitational potential energy is an energy that is linked to gravity or gravitational force.The gravitational potential energy equation is:
GPE = m × g × h,
m = mass in kilograms,
g = acceleration (9.8 ms-2 on Earth)
h = height.
Gravitational Potential Energy Derivation Equation-:
Let us consider an object, of mass M, which is placed along the x-axis, and there is a test mass m at infinity. Work done at bringing it without acceleration through a minimal distance (dx) is given by:
dw = Fdx
Here, F is an attractive force and towards the negative x-axis direction is the displacement. Therefore, F and dx are in a similar direction.
dw=(GMm/x²)dx
Now , we are Intergrating both the side of above equ,
w=∫∞r(GMm/x2)dx
w=−[GMm/x]
w=−[GMmr]−(−GMm/∞)
w=−[GMm/r]
As the potential energy is stored as U, the gravitational potential energy at ‘r’ distance from the object having mass ‘M’ is:
U = - GMm/r
Now if another mass inside the gravitational field moves from one point inside the field to another point of the field of mass M, the other mass experiences a change in potential energy given by:
ΔU = GMm (1/ri–1/rf) ( ri= initial position and rf= final position )If ri > rf then ΔU is negative.
For calculating the potential energy of earth moon system by the help of given fomula method:
U=-(G*M1*M2)/R
Where:
- U is the gravitational potential energy
- G is the gravitational constant (6.67 x 10^-11 N·m^2/kg^2)
- M1 is the mass of the Earth (5.97 x 10^24 kg)
- M2 is the mass of the Moon (7.34 x 10^22 kg)
- r is the distance between the Earth and Moon centers (3.84 x 10^8 m)
Plugging in these values gives the total gravitational potential energy of approximately 4.6 x 10^29 Joules for the Earth-Moon system.
