Vector uses in physics – Calculate velocity and acceleration – Exercise 3844

Exercise

A particle moves according to the law of motion

r(t)=cos(α)cos(ωt)i+sin(α)cos(ωt)j+sin(ωt)k\vec{r}(t)=\cos(\alpha)\cos(\omega t)\vec{i}+\sin(\alpha)\cos(\omega t)\vec{j}+\sin(\omega t)\vec{k}

Where

ω>0\omega>0

Calculate their velocity, acceleration and their values (vector sizes).

Final Answer

v(t)=ωcos(α)sin(ωt)iωsin(α)sin(ωt)j+ωcos(ωt)k\vec{v}(t)=\omega\cos(\alpha)\sin(\omega t)\vec{i}-\omega\sin(\alpha)\sin(\omega t)\vec{j}+\omega\cos(\omega t)\vec{k}

v(t)=ω|\vec{v}(t)|=\omega

a(t)=ω2cos(α)cos(ωt)iω2sin(α)cos(ωt)jω2sin(ωt)k\vec{a}(t)=-\omega^2\cos(\alpha)\cos(\omega t)\vec{i}-\omega^2\sin(\alpha)\cos(\omega t)\vec{j}-\omega^2\sin(\omega t)\vec{k}

a(t)=ω2|\vec{a}(t)|=\omega^2

Solution

Coming soon…

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