GRAVITY VERSUS THE GRAVITY PIT
In this experiment, we use an analogy of coins falling into a pit to explain how planets move around the Sun. Because the Sun’s mass, the space-time is curved around it, as a pit, making the planets orbit it. However, in the system coin-pit, different of planetary system, there are friction and air resistance that make the coin lose energy and fall in a spiral moving. Thus, to compare these two systems, we have to consider each period of the coin as unique and there are no friction and air resistance.
The planets are kept in their orbit by the force of gravity (FG) that the sun exerts on them. Where, by Newton:
where, G is the gravitational constant, M is mass of Sun, m is mass of planet and r is the distance between planet and Sun.
The force of gravity is strongest when the distance r between the Sun and the planets is smallest.
The centripetal force (FC) is , where v is the speed of the planet.
In this system FG = FC.
So, we get
That way, v is proportional to .
The force of gravity is strongest near the sun.
The period of the planets is shortest for the planets closest to the sun.
The weight (P) of the coin creates a normal force (Fn) that it is perpendicular to the surface of the pit.
This normal force has a horizontal component that is the centripetal force (FC) responsible for the orbit of coin around the center of the pit.
This centripetal force is greater when the slope of the surface is steeper, that is, when the radius (r) is smaller.
By the vector decomposition:
Fn cosα = P = mg, and
Fn sinα =
where m is the coin mass, g is gravity acceleration and v is
the coin speed. Then,
The pit has a hyperbolic superficial, where the relation between depth of the pit (y) and its radius (r) is , and k is a constant.
By the geometric of the pit: tanα
Then, we get
v is proportional to .
The slope of the gravity pit is greatest near the center.
The period of revolution of the coin is shortest when the coin is closer to center of the pit.