[Physics] Force of Gravitational attraction between the ring and the sphere?

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Let us say a uniform ring of mass m and radius 3a is kept above a sphere of mass M and radius 3a at a distance of 4a such that line joining the centres of ring and sphere is perpendicular to the plane of the ring.

So my question is what will be gravitational force of attraction between them.

I tried solving it but I am facing little problem while integrating and also that should I take sphere as point mass or simply a sphere.

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Best Answer

As per @lemon mentioned :

You can treat the sphere as a single point (this follows from point 1 of the shell theorem). So you just need to integrate along the length of the ring.

So I just did like that only.

Taking sphere as point mass then point mass will experiences one gravitational force of attraction that is:

$$df=cos\theta$$

As vertical component will cancel out.

Integrating it we will get:

$$F=\frac{GMm}{r^2}.sin\theta$$

$$F=\frac{GMm}{5a^2}.\frac{3a}{5a}$$

$$F=\frac{3GMm}{125a^2}$$

Which is our required answer and that's how we can determine the gravitational force of attraction between ring and sphere.

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