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19 September, 10:40

Derive an expression for the gravitational potential energy of a system consisting of Earth and a brick of mass m placed at Earth's center. Take the potential energy for the system with the brick placed at infinity to be zero. Express your answer in terms of or all of the variables m, mass of Earth mE, its radius RE, and gravitational constant G.

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  1. 19 September, 11:09
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    The gravitational potential energy of a system is - 3/2 (GmE) (m) / RE

    Explanation:

    Given

    mE = Mass of Earth

    RE = Radius of Earth

    G = Gravitational Constant

    Let p = The mass density of the earth is

    p = M / (4/3πRE³)

    p = 3M/4πRE³

    Taking for instance, a very thin spherical shell in the earth;

    Let r = radius

    dr = thickness

    Its volume is given by;

    dV = 4πr²dr

    Since mass = density * volume;

    It's mass would be

    dm = p * 4πr²dr

    The gravitational potential at the center due would equal;

    dV = - Gdm/r

    Substitute (p * 4πr²dr) for dm

    dV = - G (p * 4πr²dr) / r

    dV = - G (p * 4πrdr)

    The gravitational potential at the center of the earth would equal;

    V = ∫dV

    V = ∫ - G (p * 4πrdr) {RE, 0}

    V = - 4πGp∫rdr {RE, 0}

    V = - 4πGp (r²/2) {RE, 0}

    V = - 4πGp{RE²/2)

    V = - 4Gπ * 3M/4πRE³ * RE²/2

    V = - 3/2 GmE/RE

    The gravitational potential energy of the system of the earth and the brick at the center equals

    U = Vm

    U = - 3/2 GmE/RE * m

    U = - 3/2 (GmE) (m) / RE
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