Question

For the function v = 12 sin 40, calculate the: a) mean b) root mean square (RMS)
Over a range of 0 ≤ Ø ≤ π/4 radians.

(Note: the trigonometric identity cos 2Ø = 1-2sin^2 Ø)

Answer 1

a)

$$v_{av}=12\frac{4}{\pi}\int _0^{\frac{\pi}{4}}d\theta \sin{4\theta}=3\frac{4}{\pi}\int _0^{\pi}dx \sin{x}$$

$$v_{av}=-3(\cos{\pi}-\cos{0})\frac{4}{\pi}=\frac{24}{\pi}$$

b)

$$v_{rms}^2=12^2\frac{4}{\pi}\int _0^{\frac{\pi}{4}}d\theta \sin^2{4\theta}=12^2\frac{2}{\pi}\int _0^{\frac{\pi}{4}}d\theta (1-\cos{8\theta})$$

$$\int _0^{\frac{\pi}{4}}d\theta (1-\cos{8\theta})=\frac{\pi}{4}-\frac{1}{8}\int _0^{2\pi}dy \sin{y}=\frac{\pi}{4}$$

Thus,

$$v_{rms}^2=12^2\frac{4}{\pi}\int _0^{\frac{\pi}{4}}d\theta \sin^2{4\theta}=12^2\frac{2}{\pi}\frac{\pi}{4}$$

$$v_{rms}=\frac{12}{\sqrt{2}}$$