Classical Mechanics

Newton's Law of Gravity

Newton's Laws of Gravity

         

Climbing to the top of Mount Everest is hard, but it's slightly easier than you might think as people weigh less as they climb to the top. Let WEW_E be a person's weight on top of Mount Everest and WSW_S be their weight at sea level. What is the value of 1WE/WS1-W_E/W_S?

Details and assumptions

  • Assume the earth (other than Everest) is a sphere of mass 6×1024 kg6 \times 10^{24}~\mbox{kg} and radius 6,370 km6,370~\mbox{km}.
  • The top of Mount Everest is 8,848 m8,848~\mbox{m} above the surface of the earth.

If the moon moves from the opposite side of the earth from you (A in the above figure) to the facing side (B in the above figure), approximately by what percent does the moon's gravitational force on you (with mass 65 kg65 \text{ kg}) increase, assuming that the earth-moon (center-to-center) distance is 3.55×108 m3.55 \times 10^8\text{ m} and the earth's radius is 6.37×106 m?6.37 \times 10^6\text{ m}?

Assume that the earth and the moon are both perfect spheres and the earth-moon distance is constant.

If the magnitude of gravitational attraction force between two particles with respective masses 5.5 kg5.5\text{ kg} and 2.9 kg2.9\text{ kg} is 2.7×1012 N,2.7 \times 10^{-12}\text{ N}, what is the approximate separation between the two particles, given that 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 neutron star has a mass equal to that of the sun, which is 2.1×1030 kg,2.1 \times 10^{30}\text{ kg}, but has a radius of only 7 km.7\text{ km}. What is the approximate gravitational acceleration at the surface of the star?

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.

The interior structure of the Earth is layered in spherical shells, like an onion. Assume that a certain planet has similar interior structure of core and outer shell, as shown in the above figure. The radius of the core is RR and the inner and outer radii of the outer shell are RR and 2R,2R, respectively. If the mass of the core is M=4.1×1024 kgM=4.1 \times 10^{24}\text{ kg} and that of the outer shell is 4M,4M, what is the approximate gravitational acceleration of a particle which is at a distance of RR from the surface of the outer shell?

  • The universal gravitational constant is G=6.67×1011 N m2/kg2.G=6.67 \times 10^{-11} \text{ N }\cdot\text{m}^2\text{/kg}^2.
  • R=5.5×106R = 5.5\times10^6 m.
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