Q.103. Application of Integration.

Question 103.
Calculate the center of mass of a uniform solid conical volume of height h & bas radius R.

Answer 103.
Let cone be placed with its vertex at the origin, and axis along y-axis.
Considering uniform density, centre of mass will be the centre of volume.
Imagine a thin horizontal disk of thickness, dy, sliced from cone at a height y having radius x. By similar triangles,
x/y = r/h => x = (r/h)y

Volume of disk
dV = π x^2 dy = π (r/h)^2 y^2 dy
Centre of mass = centre of volume
= (1/V) ∫ (0 to h) y * [π (r/h)^2 y^2] dy
= [1 / [(1/3)π r^2 h] ] * π (r/h)^2 * [(h^4)/4]
= 3h/4
Thus centre of mass is on the axis of the cone at a distance
3h/4 from the vertex or h/4 from the base.

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