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Orbits

Believe it or not — the world does not revolve around you. Accept this harsh truth, then calculate the beguiling dance of objects in orbit, from binary stars to the symphony of our Solar System.

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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