All Problems

Mechanical Oscillations

Problem 4.6

At the moment t=0t=0 a point starts oscillating along the xx axis according to the law x=asin⁡ωt.x=a \sin \omega t . Find: (a) the mean value of its velocity vector projection ⟨vx⟩\left\langle v_{x}\right\rangle; (b) the modulus of the mean velocity vector ∣⟨v⟩∣|\langle\mathbf{v}\rangle| (c) the mean value of the velocity modulus ⟨v⟩\langle v\rangle averaged over 3/83 / 8 of the period after the start.

Reveal Answer
4.6. (a) ⟨vx⟩=223πaω\left\langle v_{x}\right\rangle=\frac{2 \sqrt{2}}{3 \pi} a \omega; (b) ∣⟨v⟩∣=223πa(0)|\langle\mathbf{v}\rangle|=\frac{2 \sqrt{2}}{3 \pi} a(0) (c) ⟨v⟩=\langle v\rangle= =2(4−2)9aω=\frac{2(4-\sqrt{2})}{9} a \omega
Prev
Next