Question

Find the center of mass of a uniform plate having semicircular inner and outer boundaries of radii R1 and R2 (Figure 9-E5).

Answer 1

Thanks for asking the question!

ANSWER::

Centre of mass of the plate will be on symmetrical axis.

Centre of mass at y

= [(πR₂²/2)(4R₂/3π) - (πR₁²/2)(4R₁/3π)] / [(πR₂²/2) - (πR₁²/2)]

= [(2/3)R₂³ - (2/3)R₁³] / [π/ 2(R₂² - R₁²)]

= [4(R₂-R₁)(R₂² + R₁² + R₁R₂)] / [3π(R₂-R₁)(R₂+R₁)]

= [4(R₂² + R₁² + R₁R₂)] / [3π (R₁ + R₂)] 

[4(R₂² + R₁² + R₁R₂)] / [3π (R₁ + R₂)] above the centre.

Hope it helps!

Answer 2

The center of mass of a uniform plate having semicircular inner and outer boundaries of radii R1 and R2 is $\frac{\left[4\left(\mathrm{R}_{2}^{2}+\mathrm{R}_{1}^{2}+\mathrm{R}_{1} \mathrm{R}_{2}\right)\right]}{\left[3 \pi\left(\mathrm{R}_{1}+\mathrm{R}_{2}\right)\right]}$ above the center.

Explanation:

Step 1:

The plate's center of mass is on the symmetrical axis.

Symmetrical axis is nothing but a line through a circle to form a mirror image on each side. The Two-half matches when the shape on the symmetry axis is folded in half .

Step 2:

Centre of mass at y  is given by

$=\frac{\frac{\pi \mathrm{R}_{2}^{2}}{2} \frac{4 \mathrm{R}_{2}}{3 \pi}-\frac{\pi \mathrm{R}_{1}^{2}}{2} \frac{4 \mathrm{R}_{1}}{3 \pi}}{\frac{\pi \mathrm{R}_{2}^{2}}{2}-\frac{\pi \mathrm{R}_{1}^{2}}{2}}$

$=\frac{\left[\left(\frac{2}{3}\right) R_{2}^{3}-\left(\frac{2}{3}\right) R_{1}^{3}\right]}{\left[\frac{\pi}{2}\left(R_{2}^{2}-R_{1}^{2}\right)\right]}$

$=\frac{\left[4\left(\mathrm{R}_{2}-\mathrm{R}_{1}\right)\left(\mathrm{R}_{2}^{2}+\mathrm{R}_{1}^{2}+\mathrm{R}_{1} \mathrm{R}_{2}\right)\right]}{\left[3 \pi\left(\mathrm{R}_{2}-\mathrm{R}_{1}\right)\left(\mathrm{R}_{2}+\mathrm{R}_{1}\right)\right]}$

$=\frac{\left[4\left(\mathrm{R}_{2}^{2}+\mathrm{R}_{1}^{2}+\mathrm{R}_{1} \mathrm{R}_{2}\right)\right]}{\left[3 \pi\left(\mathrm{R}_{1}+\mathrm{R}_{2}\right)\right]}$

$=\frac{\left[4\left(\mathrm{R}_{2}^{2}+\mathrm{R}_{1}^{2}+\mathrm{R}_{1} \mathrm{R}_{2}\right)\right]}{\left[3 \pi\left(\mathrm{R}_{1}+\mathrm{R}_{2}\right)\right]}$ above the center.