Derive the total energy (E) of a particle of mass m moving in a centripetal force field, based on its radial coordinates and derivatives. If this particle moves in a spiral orbit expressed by the equation "(@) = cơ² (c, real constant), use the energy relation you find to obtain the force F(r) that forms this orbit.
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- Consider the observation that the acceleration due to the gravitational force acting on a mass around a host planet decreases with the square of the separation between the objects. We can ask ourselves: why is it still accurate to consider a gravitational acceleration value of 9.8\frac{m}{s^2}9.8s2m for all of our projectile motion problems and all of our gravitational potential energy from prior modules? Let's analyze a situation and justify this analysis method: consider an object being launched from ground level to an altitude of 10,000 meters, roughly the cruising altitude of most jet liners, and far above our everyday experiences on Earth's surface. Compare the gravitational acceleration of the object at Earth's surface (the radius of Earth is about r_E=6.37\times10^6mrE=6.37×106m) to the acceleration value at the 10,000 meter altitude by determining the following ratio: g10,000m/gsurfaceHow much work would be required to move the moon (7.36 E 22 kg) from its current obit of 385 E 6 m to an orbit of 504 E 6 m. Assume the mass of the earth to be 6.00 E 24 kg and G = 6.67 E -11. Include the correct sign. This would be work external to the earth/moon system, not gravitational work. Think of it as the energy that would have to be added or removed from the system to allow the calculated change in gravitational potential energy.A 3500-kg spaceship is in a circular orbit 190 km above the surface of Earth. It needs to be moved into a higher circular orbit of 390 km to link up with the space station at that altitude. In this problem you can take the mass of the Earth to be 5.97 × 1024 kg. How much work, in joules, do the spaceship’s engines have to perform to move to the higher orbit? Ignore any change of mass due to fuel consumption.
- A particle is projected from the surface of the earth with a speed equal to 2.5 times the escape speed. When it is very far from the earth (i.e. infinitely far away) what is its speed?When describing the changes in energy as Felix falls, what is the system that is interacting and causing a change in potential energy? (a) Felix and the capsule he jumps from (b) Felix and the Earth (c) Felix (d) The EarthConsider two particles: p at the origin (0,0,0) = R³ with mass M > 0, and q at the point/position vector 7 = (x, y, z) = R³ with mass m > 0. Let G be the universal gravitational constant. (We will assume the MKS system of units.) The force F = F (7) felt by the particle q due to its gravitational interaction with particle p is: GMm 7(7)= == 7, for all 7 = (x, y, z) € R³\{0} . 17 Also consider the function ƒ : R³\{♂} → R given by GMm f(x, y, z) := TT , for all 7 = (x, y, z) € R³\{0} . Fix an arbitrary point/position vector = (x, y, z) in R³\{♂}. 2, calculate the (3) Calculat cade of the vector (4) Calculate the direction of the vector ₹(7). (5) Assume that is the total force on the particle q. Calculate the instantaneous acceleration, d, of the particle q when it is at the point 7 = (x, y, z).