Answered: Prove that the total gravitational… | bartleby

Related questions

Question

Prove that the total gravitational potential energy can be written as W = .5 ∫d³xρ(x)Φ(x). ρ(x) and Φ(x) are the density and gravitational potential at location x in space, respectively.

Expert Solution

Step by step

Solved in 3 steps with 11 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Give the definition of the gravitational field strength, g, in terms of the gravitational potential V
Lindsay is in an airplane, preparing to skydive. Lindsay has a mass of 71.1 kg, and the airplane is flying at an altitude of 3620 m above the ground. Before she jumps from the airplane, how much gravitational potential energy does he have? 6.44∗10^5 J 1.90∗10^6 J 2.52∗10^6 J 2.57∗10^5 J
Prove the following equation Pw(x,t)= py(x,t) with p=2 ww in (2x-1) Knowing that y (x,t)= 3e^
There exista a spherical planet with a mass of M and a radius of R. How much energy is required to take a rocket of a mass m from rest on the surface of the planet to a circular orbit a height h above the surface?Find using the Energy Difference Ef -Ei where Ef is the energy in orbit and Ei is the energy at rest on the surface. h is not small
Plan A - Get people off the Earth in a permanent space habitat. Recall that the gravitational potential energy of an object in orbit around the Earth has magnitude GMmd where M is the mass of the Earth and m is the mass of the object in orbit around it (G = 6.67×10-11Nm2/kg2). If this is also equal to the energy that must be expended (work) to place the object at orbital altitude d, i. calculate the work required to place 100 million people in orbit at 300 miles altitude. Use a mass of 70 kg for the average mass per person. (1 Joule=1 Nm) ii. How does this compare to the current annual energy consumption of the United States? (1 Watt hour = 3600 Joules)
Consider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) g. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc's centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. the gravitational potential p(2) set up by that disc is given by dr'; ()² + z² sp(2) = 27Gg
A satellite in Earth orbit has a mass of 103 kg and is at an altitude of 1.91 106 m. (Assume that U = 0 as r → ∞.) (a) What is the potential energy of the satellite–Earth system? (J) (b) What is the magnitude of the gravitational force exerted by the Earth on the satellite? (N) (c) What force, if any, does the satellite exert on the Earth? (Enter the magnitude of the force, if there is no force enter 0.) (N)
Consider 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).
A satellite has a mass of 94 kg and is located at 2.04 106 m above the surface of Earth. (a) What is the potential energy associated with the satellite at this location? J(b) What is the magnitude of the gravitational force on the satellite?