Centre of Mass of Hollow Cone

Centre of mass is the point at which the entire mass of the object is concentrated. The cone can be either a solid cone or a hollow cone. The hollow cone will have the centre of mass at a distance of 2h/3 from the apex and at a distance of h/3 from the base.

Let us determine the value of the centre of mass (centre of gravity) of a hollow cylinder.

Let us consider a hollow cone of height H. A circular ring of thickness dx is considered at a height y from the base of the cone. The radius of the ring is taken as x sinθ.

Area of the circular ring, dA = 2π(xsinθ)dx

Mass of the circular ring of thickness dx is,

dM=MπRR2+H2×2πxsinθdxdM=\frac{M}{\pi R\sqrt{R^{2}+H^{2}}}\times 2\pi xsin\theta dx dM=MRR2+H2×2xRR2+H2dxdM = \frac{M}{R\sqrt{R^{2}+H^{2}}}\times 2x\frac{R}{\sqrt{R^{2}+H^{2}}}dx

dM = 2Mxdx/(R2 +H2)——–(1)

y = H – xcosθ

y=HxHH2+R2y = H -\frac{xH}{\sqrt{H^{2}+R^{2}}} ——–(2)

Centre of mass, C=ydMdMC = \frac{\int ydM}{\int dM}——(3)

Substitute the values from equa (1) and equa (2) in equa (3) and integrate it

C=0R2+H2H(1xR2+H2)×2MR2+H2xdx0R2+H22MR2+H2xdxC =\frac{\int_{0}^{\sqrt{R^{2}+H^{2}}} H(1-\frac{x}{\sqrt{R^{2}+H^{2}}})\times \frac{2M}{R^{2}+H^{2}}xdx}{\int_{0}^{\sqrt{R^{2}+H^{2}}}\frac{2M}{R^{2}+H^{2}}xdx }

Solving the above equation we get C = H/3

Centre of Mass of the hollow cone, C = H/3

Where H is the height of the cone.


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