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Math Help - gravitation question

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    gravitation question

    a solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F1 on a particle placed at a distance 3R from the centre of the sphere a spherical cavity of radius r/2 is now made in the sphere .the sphere with cavity now applies a gravitational force F2 on the same particle.the ratio F1/F2 is
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  2. #2
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    Quote Originally Posted by prasum View Post
    a solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F1 on a particle placed at a distance 3R from the centre of the sphere a spherical cavity of radius r/2 is now made in the sphere .the sphere with cavity now applies a gravitational force F2 on the same particle.the ratio F1/F2 is
    The problem is spherically symmetric so no problems with using the usual force law for gravitation:
    \displaystyle F_1 = \frac{GMm}{(3R)^2}
    where M is the mass of the sphere and m is the mass of the object.

    To evaluate this we need the mass of the whole sphere. It has a constant mass density, call it \rho. Then we know that
    \displaystyle V = \frac{4}{3} \pi R^3 \implies M = \rho V = \frac{4}{3} \pi \rho R^3

    So
    \displaystyle F_1 = \frac{GMm}{(3R)^2} = \frac{G \left ( \frac{4}{3} \pi \rho R^3 \right ) m}{(3R)^2}

    Now your turn. Find the mass of the removed section. Notice that the problem is still spherically symmetric so that
    \displaystyle F_2 = \frac{G(M - \Delta M)m}{(3R)^2}

    where \Delta M is the removed mass.

    Now form the fraction and you will find that practically everything cancels out.

    -Dan
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