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Question

Derive the relation between linear and angular velocity,or Derive $$ v=rw $$

Solution
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Consider a body rotating about an axis passing through a point $$ O $$and perpendicular to the plane of the paper. Let $$P$$ be the position of a particle inside the body. If the body rotates through an angle $$ 0 $$ in a time interval of $$ t $$, the particle at $$P$$ reaches $$P ' $$
$$ \therefore $$ Linear displacement $$PP' =r \theta $$
$$ \therefore $$Linear velocity, $$ v=\dfrac{PP'}{t}=\dfrac{r \theta}{t } $$
But $$ \dfrac{\theta}{t}=\omega $$, the angular velocity
$$ \therefore $$Linear velocity , $$ v=r \omega $$

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