Question

A paraboloid shaped solid object is formed by rotation on parabola y=2x^2 about y axis as shown . If height of body is h then position of center of mass from origin is

Answer 1

Explanation:

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Answer 2

Answer:

Explanation:

A paraboloid shaped solid object is formed by rotation of a parabola,

$y=2x^2$

⇒$x=\sqrt{\frac{y}{2} }$

σ is the surface charge density. σ=M/A

$A=\int\limits^h_0 {2\pi x} \, dy=\int\limits^h_0 {\sqrt{y} } \, dy =\frac{4 \pi}{3\sqrt{2} }h^{\frac{3}{2} }$

dy is the element along y axis having mass dm.

The parabola is symmetric about y axis, hence the center of mass will be on the y axis.

$Y_{CM}=\frac{\int {ydm} \, }{\int dm} \, }= \frac{\int\limits^h_0 {y \sigma 2\pi x} \, dy }{M}$

⇒$\frac{1}{\sqrt{2} } \int\limits^h_0 {2\pi \sigma y^{\frac{3}{2} }} \, dx =\frac{4\pi\sigma}{\sqrt{2} *5 } h^{\frac{5}{2}$

Putting the value of A from the above we get $Y_{CM}=\frac{3}{5}h$