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Deriving Kepler's Laws

         

Two satellites each of mass \( m_1 = 3.00 \text{ kg} \) and mass \( m_2 = 6.00 \text{ kg} \) revolve around the same planet. The respective radii of their orbits are \( r_1 = 2.00 \times 10^5 \text{ m} \) and \( r_2 = 1.80 \times 10^6 \text{ m}. \) If the time period of the satellite of mass \( m_1 \) is \( T_1 = 4.00 \times 10^5 \text{ s}, \) what is that of the satellite of mass \( m_2?\)

Assumptions and Details

  • The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)

A box of mass \( m = 3 \text{ kg} \) is placed on the edge of a merry-go-round of radius \( r= 4 \text{ m}. \) The coefficient of static friction between the box and the merry-go-round is \( \mu = 0.3 . \) What is the merry-go-round's speed squared at the moment the box slides off?

Assumptions and Details

  • The gravitational acceleration is \( g= 10 \text{ m/s}^2. \)

A satellite of mass \( m = 2.00 \times 10^3 \text{ kg} \) revolves around a planet of mass \( M = 6.00 \times 10^{16} \text{ kg},\) with constant speed. If the radius of the orbit is \( r = 2.00 \times 10^5 \text{ m}, \) what is the satellite's approximate speed?

Assumptions and Details

  • The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)

A satellite of mass \( m = 5.00 \times 10^3 \text{ kg} \) revolves around a planet of mass \( M = 5.00 \times 10^{16} \text{ kg}, \) at constant speed. If the radius of the orbit is \( r = 3.00 \times 10^5 \text{ m} ,\) what is the approximate period of the revolution?

Assumptions and Details

  • The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)

Two identical satellites each of mass \( m = 6.00 \times 10 ^3 \text{ kg} \) revolve around a planet of mass \( M = 5.00 \times 10^{16} \text{ kg}. \) If the respective radii of their orbits are \( r_1 = 2.00 \times 10^5 \text{ m} \) and \( r_2 = 1.80 \times 10^6 \text{ m} ,\) what is the ratio between the speeds of the two satellites?

Assumptions and Details

  • The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)
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