Centre of mass of cone and cylinder combined
A solid rigght cylinder of radius 3cm and length 14cm and a solid circular cone of radius 3 and height 28 cm are joined together at their plane faces. The solids re made of the same uniform material. Given that the centre of gravity of the cone is a distance h/4 from its plan base where h is the height of the cone find the pos. of centre of gravity of the composite body.
Can anyone help me do this question?
As I thought about it, I d take the origin of the axis at (3,0) because the centre of mass would be on that axis so I should only find out the other coordinate. Since the centre of the cylinder is at its geometrical centre it means that its centre of mass has coordinates (3,7) and since the centre of mass of the cone its at h/4=28/4=7 it means that on the axis would be at (3,14).
Since the solids are made of the same material I can use density=mass/volume=> mass=density*volume.
Thus, taking moments I have
7*m1*g+14*m2*g=y*(m1+m2)*g
Substituting the mass and simplifying the constants I get
7*V1+14*V2=(V1+V2)*y Annd thus I find out y so the overall centre of mass would be (3,y).
For some reason this doesn t get the right answer, so can you guyys help me out, please? Thank you
$\endgroup$1 Answer
$\begingroup$Taking the centre of the circular base of the cylinder as the origin, you only need to find the vertical coordinate $\bar{y}$ as you know already that the centroid lies on the axis of symmetry.
The volume of the cylinder is $126\pi$ and the volume of the cone is $84\pi$. The density is the same throughout so we only need to consider volume $\times$ distance.
The centroid of the cylinder is at a distance $7$ from the origin.
The centroid of the cone is at a distance $14\color{red}{+7}=21$ from the origin.
So, by the principle of moments, $$126\pi\times7+84\pi\times21=(126\pi+84\pi)\times \bar{y}$$
Which leads to $$\bar{y}=12.6$$
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