Question

A rigid body is spinning about a fixed point $(3,-2,-1)$ with an angular velocity of $rad/s$, the axis of retation being in the direction of $(1,2,-2)$. Find the velocity of the particle at point $(4,1,1)$

Answer 1

MISTAKE:angular velocity is not printed

let the angular velocity be $\omega rad/s$

Given axis is passing through $(3,-2,-1)$ having d.r.'s as $(1,2,-2)$

$⟹$ equation of line is $\frac{x-3}{1}=\frac{y+2}{2}=\frac{z+1}{-2}$

let $\frac{x-3}{1}=\frac{y+2}{2}=\frac{z+1}{-2}=t⟹x=3+t,y=-2+2t,z=-1-2t$

any point lies on the axis will be $(3+t,-2+2t,-1-2t)$

for a point on the axis to be perpendicular to $(4,1,1)$,$\left(1\right)(3+t-4)+\left(2\right)(-2+2t-1)+(-2)(-1-2t-1)=0$

$⟹t-1+4t-6+4t+4=0$

$t=\frac{1}{3}$

perpendicular point is $(\frac{10}{3},\frac{-4}{3},\frac{-5}{3})$

perpendicular distance from $(4,1,1)=\sqrt{(\frac{-2}{3}{)}^2+(\frac{7}{3}{)}^2+(\frac{8}{3}{)}^2}=\frac{\sqrt{117}}{3}$

velocity of particle $=($perpendicular distance from axis$)\times$angular velocity$=\frac{\sqrt{117}\omega }{3}$