All Problems Mechanical Oscillations
Problem 4.6
At the moment \(t=0\) a point starts oscillating along the \(x\) axis according to the law \(x=a \sin \omega t .\) Find: (a) the mean value of its velocity vector projection \(\left\langle v_{x}\right\rangle\); (b) the modulus of the mean velocity vector \(|\langle\mathbf{v}\rangle|\) (c) the mean value of the velocity modulus \(\langle v\rangle\) averaged over \(3 / 8\) of the period after the start.
4.6. (a) \(\left\langle v_{x}\right\rangle=\frac{2 \sqrt{2}}{3 \pi} a \omega\); (b) \(|\langle\mathbf{v}\rangle|=\frac{2 \sqrt{2}}{3 \pi} a(0)\) (c) \(\langle v\rangle=\) \(=\frac{2(4-\sqrt{2})}{9} a \omega\)