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- Homework Statement
- Find the center of mass of a uniform this semicircular plate of radius R. Let the origin be at the center of the semicircle, the plate arc from the +x axis to the -x axis, and the z axis be perpendicular to the plate.

- Relevant Equations
- ##rcm = \frac 1 M*\int(r*dm)##

$$rcm = \frac{1}{M}\int_0^\pi(rdm)$$

$$dm = \sigma{dA}$$

$$dA = (\pi*R^2)*\frac{d\theta}{2\pi}$$

$$\sigma = \frac{M}{\frac{\pi*R^2}{2}}$$

$$dm = M*\frac{d\theta}{\pi}$$

$$r = R(cos(\theta)\vec i + sin(\theta)\vec j)$$

$$rcm = \int_0^\pi{\frac{R}{\pi}(cos(\theta)\vec i + sin(\theta)\vec j)} = \frac{2*R}{\pi}\vec j$$

Where did I go wrong?

$$dm = \sigma{dA}$$

$$dA = (\pi*R^2)*\frac{d\theta}{2\pi}$$

$$\sigma = \frac{M}{\frac{\pi*R^2}{2}}$$

$$dm = M*\frac{d\theta}{\pi}$$

$$r = R(cos(\theta)\vec i + sin(\theta)\vec j)$$

$$rcm = \int_0^\pi{\frac{R}{\pi}(cos(\theta)\vec i + sin(\theta)\vec j)} = \frac{2*R}{\pi}\vec j$$

Where did I go wrong?

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