GRAVITATIONAL ENERGY: FORMULAS, CHARACTERISTICS, APPLICATIONS, EXERCISES - PHYSICAL - 2024

The gravitational energy is having a massive object when it is immersed in the gravitational field produced by another. Some examples of objects with gravitational energy are: the apple on the tree, the falling apple, the Moon orbiting the Earth, and the Earth orbiting the Sun.

Isaac Newton (1642-1727) was the first to realize that gravity is a universal phenomenon and that every object with mass in its environment produces a field capable of producing a force on another.

Figure 1. The Moon orbiting the Earth has gravitational energy. Source: Pixabay

Formulas and equations

The force Newton was referring to is known as the gravitational force and provides energy to the object on which it acts. Newton formulated the law of universal gravitation as follows:

"Let there be two point objects of masses m1 and m2 respectively, each one exerts an attractive force on the other that is proportional to the product of their masses and inversely proportional to the square of the distance that separates them."

The gravitational energy U associated with the gravitational force F is:

An object that is immersed in a gravitational field has gravitational potential energy U and kinetic energy K. If there are no other interactions, or they are of negligible intensity, the total energy E of said object is the sum of its gravitational energy plus its kinetic energy:

E = K + U

If an object is in a gravitational field and no other dissipative forces are present, such as friction or air resistance, then the total energy E is a quantity that remains constant during motion.

Characteristics of gravitational energy

- An object has gravitational potential energy if it is only in the presence of the gravitational field produced by another.

- The gravitational energy between two objects increases as the separation distance between them is greater.

- The work done by the gravitational force is equal to and contrary to the variation of the gravitational energy of the final position with respect to that of its initial position.

- If a body is subjected only to the action of gravity, then the variation of its gravitational energy is equal to and contrary to the variation of its kinetic energy.

- The potential energy of an object of mass m that is at a height h with respect to the earth's surface is mgh times greater than the potential energy at the surface, where g is the acceleration of gravity, for heights h much less than the earth's radius.

Gravitational field and potential

The gravitational field g is defined as the gravitational force F per unit mass. It is determined by placing a test particle m at each point in space and calculating the quotient between the force acting on the test particle divided by the value of its mass:

g = F / m

The gravitational potential V of an object of mass m is defined as the gravitational potential energy of that object divided by its own mass.

The advantage of this definition is that the gravitational potential only depends on the gravitational field, so that once the potential V is known, the gravitational energy U of an object of mass m is:

U = mV

Figure 2. Gravitational field (solid lines) and equi-potentials (segmented line) for the Earth - Moon system. Source: WT Scott, Am. J. Phys. 33, (1965).

Applications

Gravitational potential energy is what bodies store when they are in a gravitational field.

For example, the water contained in a tank has more energy as the tank is higher.

The higher the tank height, the greater the speed of the water leaving the tap. This is due to the fact that the potential energy of the water at the height of the tank is transformed into kinetic energy of the water at the outlet of the tap.

When water is dammed high on a mountain, that potential energy can be harnessed to turn power generation turbines.

Gravitational energy also explains the tides. Since the energy and gravitational force depend on distance, the gravitational pull of the Moon is greater on the face of the Earth closest to the Moon than the face farthest and opposite.

This produces a difference in forces that deforms the surface of the sea. The effect is greatest at a new moon, when the Sun and Moon are aligned.

The possibility of building space stations and satellites that remain relatively close to our planet is due to gravitational energy produced by the Earth. Otherwise, space stations and artificial satellites would be roaming through space.

Earth's gravitational potential

Suppose that the Earth has mass M and an object that is above the earth's surface at a distance r from its center has mass m.

In this case the gravitational potential is determined from the gravitational energy simply dividing by the mass of the object resulting:

Potential energy near the earth's surface

Suppose the Earth has radius R _T and mass M.

Even when the Earth is not a point object, the field on its surface is equivalent to that which would be obtained if all its mass M were concentrated in the center, so that the gravitational energy of an object at height h above the Earth's surface is

U (R _T + h) = -GM m (R _T + h) ^ - 1

But because h is much less than R _T, the above expression can be approximated by

U = Uo + mgh

Where g is the acceleration of gravity, whose average value for Earth is 9.81 m / s ^ 2.

Then the potential energy Ep of an object of mass m at height h above the earth's surface is:

Ep (h) = U + Uo = mgh

On the Earth's surface h = 0, so an object on the surface has Ep = 0. Detailed calculations can be seen in Figure 3.

Figure 3. Gravitational potential energy at a height h above the surface. Source: prepared by F. Zapata.

Exercises

Exercise 1: Gravitational collapse of the Earth

Suppose that our planet undergoes a gravitational collapse due to loss of thermal energy in its interior and its radius falls to half its current value but the mass of the planet remains constant.

Determine what the acceleration of gravity near the surface of New Earth would be and how much a survivor weighing 50 kg-f would weigh before collapse. Increase or decrease the gravitational energy of the person and by what factor.

Solution

The acceleration of gravity on the surface of a planet depends on its mass and its radius. The constant of gravitation is universal and works equally for planets and exoplanets.

In the present case, if the radius of the Earth is reduced by half then the acceleration of gravity of the New Earth would be 4 times greater. Details can be seen on the board below.

This means that a superman and survivor who weighed 50 kg-f on the old planet will weigh 200 kg-f on the new planet.

On the other hand, the gravitational energy will have been halved on the surface of the new planet.

Exercise 2: Gravitational Collapse and Escape Velocity

In reference to the situation presented in exercise 1, what would happen to the escape velocity: it increases, it decreases, by what factor?

Solution 2

Escape velocity is the minimum velocity necessary to escape the gravitational pull of a planet.

To calculate it, it is assumed that a projectile that is fired with this speed reaches infinity with zero speed. Furthermore, at infinity the gravitational energy is zero. Therefore a projectile fired with escape velocity will have zero total energy.

That is to say that on the surface of the planet at the time of the shot the sum of the kinetic energy of the projectile + the gravitational energy must be zero:

½ m Ve ^ 2 - (G Mm) / R _T = 0

Note that the escape velocity does not depend on the mass of the projectile and its value squared is

Ve ^ 2 = (2G M) / R _T

If the planet collapses to a radius half of the original, the square of the new escape velocity becomes double.

Therefore the new escape velocity grows and becomes 1.41 times the old escape velocity:

Go '= 1.41 Go

Exercise 3: Gravitational energy of the apple

A boy on the balcony of a building 30 meters above the ground drops a 250 g apple, which after a few seconds reaches the ground.

Figure 4. As it falls, the potential energy of the apple is transformed into kinetic energy. Source: PIxabay.

a) What is the gravitational energy difference of the apple at the top with respect to the apple at ground level?

b) How fast was the apple just before spilling onto the ground?

c) What happens to the energy once the apple is flattened against the ground?

Solution

a) The gravitational energy difference is

mgh = 0.250 kg * 9.81 m / s ^ 2 * 30 m = 73.6 J

b) The potential energy that the apple had when it was 30 m high is transformed into kinetic energy by the time the apple reaches the ground.

½ mv ^ 2 = mgh

v ^ 2 = 2.gh

By substituting values ​​and solving, it follows that the apple reaches the ground with a speed of 24.3 m / s = 87.3 km / h.

c) Obviously the apple is scattered and all the gravitational energy accumulated at the beginning is lost in the form of heat, since the pieces of apple and the impact zone heat up, in addition part of the energy is also dissipated in the form of sound waves " splash ".

References

1. Alonso, M. (1970). Physics Vol. 1, Inter-American Educational Fund.
2. Hewitt, Paul. 2012. Conceptual Physical Science. 5th. Ed. Pearson.
3. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
4. Sears, F. (2009). University Physics Vol. 1
5. Wikipedia. Gravitational energy. Recovered from: es.wikipedia.com
6. Wikipedia. Gravitational energy. Recovered from: en.wikipedia.com