Question 747: Centre of Mass and Rotational Motion

Question

Two vectors $$\overrightarrow A $$  and $$\overrightarrow B $$ are defined as $$\overrightarrow A $$ $$=$$ $$a\widehat i$$ and $$\overrightarrow B  = a$$ $$\left( {\cos \,\omega T\widehat i + \sin \,\omega t\,\widehat j} ight),$$ where $$a$$ is a constant and $$\omega  = \pi /6\,\,rad{s^{ - 1}}.$$  If $$\left| {\overrightarrow A  + \overrightarrow B } ight| = \sqrt 3 \left| {\overrightarrow A  - \overrightarrow B } ight|$$ at time $$t = 	au $$ for the first time, the value of $$	au ,$$ in
second,
is ______________.

Accepted Answer

The vector $$\overrightarrow A  = \alpha \widehat i$$ has a magnitude a and its direction is fixed. The vector $$\overrightarrow B  = a(\cos \omega t\widehat i + \sin \omega t\widehat j)$$ rotates with an angular speed $$\omega$$ = $$\pi$$ / 6 rad/s in a circle of radius a. The magnitude of the sum and the difference of these vectors are given by
$$\left| {\overrightarrow A  + \overrightarrow B } ight| = \left| {(a + a\cos \omega t)\widehat i + a\sin \omega t\widehat j} ight|$$
$$ = a\sqrt {{{(1 + \cos \omega t)}^2} + {{\sin }^2}\omega t} $$
$$ = 2a\cos (\omega t/2)$$.
$$\left| {\overrightarrow A  - \overrightarrow B } ight| = \left| {(a - a\cos \omega t)\widehat i - a\sin \omega t\widehat j} ight|$$
$$ = 2a\sin (\omega t/2)$$
Hence,
$${{\left| {\overrightarrow A  - \overrightarrow B } ight|} \over {\left| {\overrightarrow A  + \overrightarrow B } ight|}} = 	an {{\omega t} \over 2} = {1 \over {\sqrt 3 }}$$,
which gives $$t = 	au  = (2/\omega )(\pi /6) = 2$$ s.

Centre of Mass and Rotational Motion

Integer / Numeric Answer Type