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+1 vote
3.5k views
in Physics by (40 points)
edited by

Distance of the centre of mass of a solid uniform cone from its vertex is z0. If the radius of its base is R and its height is h, then z0 is equal to

(a) h2/4R 

(b) 3h/4 

(c) 5h/8 

(d) 3h2/8R

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1 Answer

0 votes
by (323k points)

The correct option is (b) 3h/4.

Explanation:

Suppose the cylindrical symmetry of the problem to note that the center of mass must lie along the z axis (x = y = 0). The only issue is how high does it lie. If the uniform density of the cone is ρ , then first compute the mass of the cone. If we slice the cone into circular disks of area πrand height dz, the mass is given by the integral:

However, we know that the radius r starts at a for z = 0, and goes linearly to zero when z = h . This means that, r = a (1-z/h), so that:

As a result, the center of mass of the cone is along the symmetry axis, one quarter of the way up from the base to the tip and 3/4 h from the tip.

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