Question

Prove the relation, v = w × r, where symbols have their usual meanings​

Answer 1

Correct Question:

In uniform circular motion, prove the relation $\large\rm { \vec{v} = \vec{ \omega} \times \vec{r}}$ where symbols have their usual meanings.

Answer:

Consider a particle revolving in the anticlockwise sense along the circumference of a circle of radius r with center O

$\large\rm { \vec{v} =}$ Linear velocity of particle

$\large\rm { \vec{ \omega} = }$ angular velocity of particle

$\large\rm { \vec{r}=}$ radius vector of particle

In the vector from, the linear displacement is

$\large\rm { \vec { \delta} = \vec{ \delta \theta} \times \vec{r}}$

dividing both sides by $\large\rm { \delta t}$ we get,

$\large\rm { \frac{ \vec{ \delta s}}{ \vec{ \delta t}} = \frac{ \vec{ \delta \theta}}{ \vec{ \delta t}} \times \vec{r}}$

$\rm { \displaystyle\lim_{ \delta t \to 0} \frac{ \vec{ \delta s}}{ \vec{ \delta t}} = \displaystyle\lim_{ \delta t \to 0} \frac{ \vec{ \delta \theta}}{ \vec{ \delta t}} \times \vec{r}}$

$\large\rm { \therefore \frac{ \vec{ \delta s}}{ \vec{ \delta t}} = \frac{ \vec{ \delta \theta}}{ \vec{ \delta t}} \times \vec{r}}$

we know that $\large\rm { \frac{ \vec{ \delta s}}{ \vec{ \delta t}} = \vec{v}}$

and $\large\rm { \frac{ \vec{ \delta \theta}}{ \vec{ \delta t}} = \omega}$

$\large\boxed{\rm{ \therefore \vec{v} = \vec{ \omega} \times \vec{r}}}$