Question

Physics:: Gravitation

Answer 1

We've been given a Sphere and a one dimensional rod.
So, now we have to calculate the force between the uniform sphere and the rod

Since, the sphere is purely symmetric, we can consider that the whole mass of the sphere lies at it's centre.

So, we have to now calculate the force between that particle and rod.

Now, you see that the distance between the particle and every small mass element of the rod, the distance is different, so the force between each mass element of the rod and the particle is different, that means there's gonna be the use of integration at some point in the sum

Alright , let's start with the solution

Let's first calculate the force between a small mass element namely 'dm' of the rod

The mass of this dm element
= m/l * dx
Since the length of that small element is very small and ultimately dm.

Now, since the variable here, is the distance from particle M to each element of the rod dm

Let's take the distance to be 'x'

Now,
Small force due to each small element
= G*M*dm/x^2
= GMm/l * dx/x^2

Let's integrate this so as to get the total force

Total force = GmM/l * Integration of dx/x^2
Now, let's see the limits of x here.

Be very careful, the limits must be r to r+l
because the only variable here was the distance between one end of the rod to other end of the rod

So we have to take those distances as the limits

So,
Integration limit for x is from r to r+l
Integration of 1/x^2 dx = -1/x
Now,
Total force = GmM/l * [ -1/x] from r to r+l

Total force = GMm/l [ -1/r - (-1/r+l)]
Simplifying further,

We get,

Total force = GMm/r(r+l)

So here option (A) is correct answer.

Hope this helps you !

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