Therefore, the gravitational force

**that the Earth exerts on the Moon is perpendicular to Moon's velocity**

*F*_{g}**, so it is a centripetal force**

*v***, making the trajectory of the Moon bend:**

*F*_{c}$$ F_{g}=F_{c} \\

\frac{GMm}{r^2}=\frac{mv^2}{r} $$ where

*G*is Newton's constant,

*M*is Earth's mass,

*m*is Moon's mass and

*r*is the radius of the orbit.

This implies that the kinetic energy of the Moon is

$$

K=\frac{1}{2}mv^2=\frac{GMm}{2r}

$$ which is smaller than the absolute value of the potential energy

$$

U=-\frac{GMm}{r}

$$ So the mechanical energy of the Moon is

$$

E=-\frac{GMm}{2r}

$$

We know that at the time of its formation, the Moon sat much closer to the Earth, a mere 22,500 km away, compared with the 402,336 km between the Earth and the Moon today. So the Moon is getting further away from Earth, now at the rate of 3.78 cm per year. Nevertheless, according to the last equation, a larger

*r*means that the Moon has more energy every year. Is its energy non conserved? Who is giving energy to the Moon?

La energía podría proceder de los fotones emitidos por el Sol, los cuales tienen una cierta energía cinética, que chocan contra la Luna, siendo un efecto similar al de la vela solar.

ReplyDeleteGood attempt. I will give you a clue: the growth-bands of fossil corals and shellfish from the Devonian and Pennsylvanian will lead you to the answer.

DeleteBuena pista. Podría ocurrir que la Luna, por atracción gravitatoria produzca movimientos en la Tierra que la ralenticen en su rotación. Como en el sistema Tierra-Luna, despreciando la energía que les llega en forma de radiación, se conserva la energía, si la Tierra pierde energía, la Luna tendrá que ganarla.

ReplyDeleteVery good!

DeleteThe migration of the Moon away from the Earth is mainly due to the action of the Earth's tides. The Moon is kept in orbit by the gravitational force that the Earth exerts on it, but the Moon also exerts a gravitational force on our planet.

Because the side of the Earth that faces the Moon is closer, it feels a stronger pull of gravity than the center of the Earth. Similarly, the part of the Earth facing away from the Moon feels less gravity than the center of the Earth. This effect stretches the Earth a bit, making it a little bit oblong. We call the parts that stick out "tidal bulges". The actual solid body of the Earth is distorted a few centimeters, but the most well-known effect is the tides raised on the ocean.

Because the Earth rotates faster (once every 24 hours) than the Moon orbits (once every 27.3 days), the bulge tries to speed up the Moon, and pull it ahead in its orbit. Because of the previous formula of the centripetal force, the more speed the Moon has, the more radius, so the Moon is pushed into a larger radius orbit. On the other hand, the Moon is also pulling back on the tidal bulge of the Earth, slowing the Earth's rotation.

In terms of energy, we can say that some of the energy of the spinning Earth gets transferred to the tidal bulge via friction. By attraction, the tidal bulge feeds a small amount of energy into the Moon, pushing it into a higher orbit. So the Moon is going to a higher orbit by taking energy from the rotation energy of the Earth!

See this link for more information.

This cognitive conflict is suitable in order to introduce the concepts of rotation energy of a rigid body and of tidal gravitational forces in senior year (12th grade).