Water flowing from a sphere

Calculus Level 5

A hollow sphere of mass \(m\) and radius \(1\) metre is placed with it's centre at origin in \(xyz\)-space. It is filled completely with water of mass \(m\). So, now the total mass of system is \(2m\).

A small hole is made at \(t=0\) at the bottom and water starts flowing out of the sphere with a constant volume flow rate.

If the maximum distance of centre of mass of the system from origin during the time \(t=0\) to the time water completely flows out from the sphere is \({a}^{\frac{1}{n}}-{b}^{\frac{1}{n}} \) meters, then what is the value of \(a+b+n?\)

Details and assumptions

  • \(a,b,n\) are distinct integers such that \(0<a,b,n<20\).

  • While calculating centre of mass of the system don't consider the water that flown out from the sphere. Only consider the water that is present inside the sphere.


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