Deriving Kepler's Laws

         

Two satellites each of mass m1=3.00 kg m_1 = 3.00 \text{ kg} and mass m2=6.00 kg m_2 = 6.00 \text{ kg} revolve around the same planet. The respective radii of their orbits are r1=2.00×105 m r_1 = 2.00 \times 10^5 \text{ m} and r2=1.80×106 m. r_2 = 1.80 \times 10^6 \text{ m}. If the time period of the satellite of mass m1 m_1 is T1=4.00×105 s, T_1 = 4.00 \times 10^5 \text{ s}, what is that of the satellite of mass m2? m_2?

Assumptions and Details

  • The universal gravitational constant is G=6.67×1011 Nm2/kg2.G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.

A box of mass m=3 kg m = 3 \text{ kg} is placed on the edge of a merry-go-round of radius r=4 m. r= 4 \text{ m}. The coefficient of static friction between the box and the merry-go-round is μ=0.3. \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 m/s2. g= 10 \text{ m/s}^2.

A satellite of mass m=2.00×103 kg m = 2.00 \times 10^3 \text{ kg} revolves around a planet of mass M=6.00×1016 kg, M = 6.00 \times 10^{16} \text{ kg}, with constant speed. If the radius of the orbit is r=2.00×105 m, r = 2.00 \times 10^5 \text{ m}, what is the satellite's approximate speed?

Assumptions and Details

  • The universal gravitational constant is G=6.67×1011 Nm2/kg2.G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.

A satellite of mass m=5.00×103 kg m = 5.00 \times 10^3 \text{ kg} revolves around a planet of mass M=5.00×1016 kg, M = 5.00 \times 10^{16} \text{ kg}, at constant speed. If the radius of the orbit is r=3.00×105 m, 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=6.67×1011 Nm2/kg2.G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.

Two identical satellites each of mass m=6.00×103 kg m = 6.00 \times 10 ^3 \text{ kg} revolve around a planet of mass M=5.00×1016 kg. M = 5.00 \times 10^{16} \text{ kg}. If the respective radii of their orbits are r1=2.00×105 m r_1 = 2.00 \times 10^5 \text{ m} and r2=1.80×106 m, 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=6.67×1011 Nm2/kg2.G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.
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