Derive Center of gravity of hemi-spheree symmetrical about Y-axis
Answers
Answer:
Consider an element disc of radius r and thickness dx at a distance x from the point O.
Then r=xtanα and volume of the disc =πx2tan2αdx
Hence, its mass dm=πx2tanαdx⋅ρ (where ρ= density of the cone
=31πR2hm)
Moment of inertia of this element, about the axis OA,
dI=dm2r2
=(πx2tan2αdx)2x2tan2x
=2πρx4tan4αdx
Thus the sought moment of inertia =2πρtan4α∫0hx4dx
=10h4πρR4⋅h5(astanα=hR)
Hence I=103mR
Explanation:
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.
Isaac Newton proved the shell theorem[1] and stated that:
A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.
If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell.
A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass.