v-2i-2j-5k-r-i-9j-8k-Find-I-can-do-r-v-r-2-and-i-get-61i-21j-16k-146-but-i-dont-get-w-r-v-why-

Question Number 16401 by sushmitak last updated on 21/Jun/17

v = 2 i + 2 j + 5 k

r = i + 9 j - 8 k

Find ω

I can do \frac{r \times v}{r^{2}} = ω

and i get ω = \frac{61 i - 21 j - 16 k}{146}

but i dont get w \times r = v .

w h y ?

Answered by ajfour last updated on 21/Jun/17

\bar{v} = \bar{ω} \times \bar{r} (a g r e e d)

\bar{r} \times \bar{v} = \bar{r} \times (\bar{ω} \times \bar{r})

= r^{2} \bar{ω} - \bar{r} (\bar{ω} . \bar{r})

I f \bar{ω} . \bar{r} \neq 0, \bar{r} \times \bar{v} \neq r^{2} \bar{ω} .

Commented by prakash jain last updated on 21/Jun/17

v = w \times r

This gives the component of v

which is perpendicular to r .

In the given question

v \cdot r = (2 + 18 - 40) = - 20 \neq 0

component of v parallel to r

= \frac{v \cdot r}{r^{2}} r = - \frac{20}{1^{1} + 9^{2} + {(- 8)}^{2}} (i + 9 j - 8 k)

= - \frac{20}{146} (i + 9 j - 8 k)

comoponent of v perpendicular

to r = v - \frac{v \cdot r}{r^{2}} r

= (2 i + 2 i + 5 k) + (\frac{20}{146} i + \frac{180}{146} j - \frac{160}{146} k)

= \frac{312 i + 472 j + 570 k}{140}

- - - - -

v = ω \times r = \frac{1}{146} (61 i - 21 j - 16 k) \times (i + 9 j - 8 k)

= \frac{1}{146} (312 i + 472 j + 570 k)

which is the expected result .

Also note that only tangential

component of velocity contributes

to angular velocity .