Finding the Centre of Mass of a Solid Cone

[AdityaDev]

Homework Statement

Find the centre of mass of solid cone.

Homework Equations

$$y_{cm}=\frac{1}{M}\int_0^Hydm$$

The Attempt at a Solution

First I took thin disks. I got the answer when I assumed its thickness to be dy but then dysecθ would be more accurate if half angle of cone is θ since dysecθ gives the length of the slanted part, while dy would give a cylinder.

Now I get wrong answer.

 

[ehild]

Homework Statement

Find the centre of mass of solid cone.

Homework Equations

$$y_{cm}=\frac{1}{M}\int_0^Hydm$$

The Attempt at a Solution

First I took thin disks. I got the answer when I assumed its thickness to be dy but then dysecθ would be more accurate if half angle of cone is θ since dysecθ gives the length of the slanted part, while dy would give a cylinder.

But the volume of a truncated cone is (h pi/3 )(r₁²+r₁r₂+r₂²) where r₁ and r₂ are the radii of the base plates and h is the height. In case of the volume element, r₁≈r₂=r and h=dy, so the volume element is that of a disk with radius r and height dy.

Also, you have to use density times Ady instead of dm in the integral.

 

[AdityaDev]

But the volume of a truncated cone is (h pi/3 )(r₁²+r₁r₂+r₂²) where r₁ and r₂ are the radii of the base plates and h is the height. In case of the volume element, r₁≈r₂=r and h=dy, so the volume element is that of a disk with radius r and height dy.

Also, you have to use density times dy instead of dm in the integral.

OK. But what's wrong in dysecθ?

 

[ehild]

Nothing is wrong with it if you do not use it instead of dy.