Doubt & Answer Explanation

Answer Explanation

Centre of mass of solid cone:



The centre of mass is the sum of the product of their masses and their displacement vectors all divided by the sum of the masses.


Consider a disc element taken anywhere inside the cone with the face of the disc parallel to the x-y plane.

dm = k * ?(z*R/h)² * dz where k is the mass per unit volume of the cone.

z*r/h is the radius of the elemental disc at a distance z from the tip of the cone.


dm*z = ?(z*R/h)² * dz * k * z where z is the position of the centre of mass of the elemental disc.


Therefore centre of mass = (1/M)(??(z*R/h)² * dz * k * z) = (1/M)(?R²/h²)k(?z³dz) with limits from z = 0 to z = h.

This comes out to be (1/M)(?R²/h²)k(h⁴/4). k =  3M/?R²h.

Therefore, centre of mass is at 3h/4.

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Now follow a similar procedure for the centre of mass of a hollow cone.

In this case, you can consider ring elements throughout the cone instead of disc elements.

dm = k * 2?(z*R/h) * dz where k is the mass per unit area of the cone. 


z*r/h is the radius of the elemental disc at a distance z from the tip of the cone.



dm*z = 
k * 2?(z*R/h) * dz * z where z is the position of the centre of mass of the elemental disc. 


Therefore centre of mass = (1/M)(?
k * 2?(z*R/h) * dz * z ) = (2k/M)(?R/h)(?z²dz) with limits from z = 0 to z = h. 


This comes out to be (2k/M)(?R/h)(h³/3). Substituting the value of k we get the centre of mass as 2h/3.


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The centre of mass of a cylinder - hollow or solid is at a distance h/2 from the base because of symmetry.