Mechanics: Gravity Pit / Newton's Cradle / Pulleys / Spinning Platform / Gyroscope / Foucault Pendulum / Torsion Pendulum / Unequal Arm Balance / Coupled Pendulums /

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.

Pit1
The similarity:

  • The planets are kept in their orbit by the force of gravity (FG) that the sun exerts on them. Where, by Newton:    

FNewton,

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.



Solar1

  • The centripetal force (FC) is Fc, where v is the speed of the planet.

  

  

  • In this system  FG = FC.

  

  • So, we get vG ,

.

  • That way, v is proportional to invR .

  

As conclusion:

  • The force of gravity is strongest near the sun.

  • The period of the planets is shortest for the planets closest to the sun.

  • The speed of the planets is greatest for the planets that revolve 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.

Pit3

Pit Surface

  • By the vector decomposition:

  • Fn cosα = P = mg, and

  • Fn sinα = Fc where m is the coin mass, g is gravity acceleration and v is the coin speed. Then, tanα =

  • The pit has a hyperbolic superficial, where the relation between depth of the pit (y) and its radius (r) is  hyperb , and k is a constant.

  • By the geometric of the pit: tanα tan2

  • Then, we get vP

  • That way, v is proportional to invR.

    As conclusion:

  • 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.

  • The speed of the coin is greatest when the coin is   near the center.
See the Link:

The history of Gravity