Question

1. what is keplers 3rd law of planetary motion?
2. define orbital period.

materials:

• string - straw - 2 washers

procedures:
a. get a piece of string and thread it through a straw.
b. tie a washer to each end of the string.
c. hold the straw. pull the string through the straw so most of the string is facing away from you.
d. swing this longer side of the string/washer in a circular motion at a consistent speed. (figure 1)
e. as you continue swinging the string, slowly pull the washer on the shorter end of the string (figure 2). observe what happens to the speed and motion of the rotating washer.

1. what happened to the rotating washer when you pulled on the opposite end of the string?
2. how does the distance from the sun affect a planets orbital velocity?
3. based on this information, which planet in our solar system has the fastest orbital velocity? which planet has the slowest orbital velocity?

Explanation:

1. Kepler's 3rd law of planetary motion states that the square of the orbital period \(T\) of a planet is proportional to the cube of the semi - major axis \(a\) of its orbit. Mathematically, \(T^{2}\propto a^{3}\) or \(T^{2}=\frac{4\pi^{2}}{GM}a^{3}\) (where \(G\) is the gravitational constant and \(M\) is the mass of the central body, e.g., the Sun).
2. The orbital period is the time taken for an object (e.g., a planet) to complete one full orbit around a central body (e.g., the Sun).
3. When you pulled on the opposite end of the string, the rotating washer sped up. This is because by reducing the radius of the circular motion (pulling the string), angular momentum \(L = mvr\) (where \(m\) is mass, \(v\) is velocity, and \(r\) is radius) is conserved. If \(r\) decreases and \(m\) is constant, \(v\) must increase.
4. According to Kepler's laws and the conservation of angular momentum, as the distance from the Sun (semi - major axis \(a\) or radius \(r\)) increases, the orbital velocity \(v\) of a planet decreases. This is because \(v=\sqrt{\frac{GM}{r}}\) (derived from the balance of gravitational and centripetal forces \(F_{g}=\frac{GMm}{r^{2}}\) and \(F_{c}=\frac{mv^{2}}{r}\)).
5. Mercury has the fastest orbital velocity because it is the closest planet to the Sun. Neptune (or Pluto, but Pluto is now classified as a dwarf planet) has one of the slowest orbital velocities among the major planets as it is far from the Sun.

Answer:

1. The square of the orbital period of a planet is proportional to the cube of the semi - major axis of its orbit (\(T^{2}\propto a^{3}\)).
2. The time taken for an object to complete one full orbit around a central body.
3. The rotating washer sped up.
4. As the distance from the Sun increases, the orbital velocity of a planet decreases.
5. Mercury has the fastest orbital velocity. Neptune has one of the slowest orbital velocities.