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Classical Mechanics

Newton's Law of Gravity

Gravitational Potential Energy

         

What is the approximate gravitational potential energy of the two-particle system, of masses \(5.4\text{ kg}\) and \(2.9 \text{ kg},\) separated by a distance of \(15.0\text{ m}?\)

Assumptions and details

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

Consider an \(xy\)-plane in deep space, where two particles A and B are located. Particle A is fixed at the origin and particle B is apart from particle A with a distance of \(3.6\text{ m}.\) The masses of particles A and B are \(24\text{ kg}\) and \(12\text{ kg},\) respectively. If particle B is released from rest, what is the kinetic energy of B when it has moved \(0.1\text{ m}\) toward A?

Assumptions and Details

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

The mean diameter of Mars is \(6.7\times 10^3\text{ km}\) and its mass is \(6.7\times 10^{23}\text{ kg}.\) What is the approximate escape speed on Mars?

Assumptions and Details

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

Four identical particles are forming a square with side length \(12.0\text{ m},\) as shown in the above figure. Then the side length of the square is reduced to \(4.0\text{ m}.\) If the mass of each particles is \(19.0\text{ kg},\) approximately how much gravitational potential energy of the four-particle system is reduced, assuming that the four particles are point particles?

Assumptions and Details

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

If a \(6000 \text{ kg}\) space rocket is launched vertically from the surface of the Earth with initial energy \(7.0 \times 10^{10} \text{ J},\) what will be the kinetic energy of the space rocket when it is \(1.4 \times 10^7 \text{ m}\) from the center of the Earth, assuming that the mass and the radius of the Earth are \(6.0 \times 10^{24}\text{ kg}\) and \(6.4 \times 10^6 \text{ m},\) respectively?

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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