21Ohiort'm
a rigid body rotates with constant angular velocity w about the line whose vector equation is r=lambda(i+2j+2k). the speed of the particle at the instant it passes through the point with p.v. 2i+3j=5k is
a) w√2 b) 2w c)w/√2 d)none
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2 Answers
30Since, the angular velocity is constant,
Speed at any point = \omega r
The radius will be the shortest distance from the line to the point (2,3,5).
Let the normal vector(to the axis of rotation) joining (2,3,5) and a point on the line be denoted by d.
\vec{d}=\left( 2-\lambda \right)\hat{i}+\left(3-2\lambda \right)\hat{j}+\left(5-2\lambda \right)\hat{k}
Now \vec{d} will be perpendicular to \left( \hat{i}+2\hat{j}+2\hat{k}\right).
\Rightarrow \left( 2-\lambda\right) +2\left(3-2\lambda \right) + 2\left(5-2\lambda \right) =0
\Rightarrow \boxed{\lambda =2}
\vec{d}=-\hat{j}+\hat{k}\Rightarrow \boxed{|\, \vec{d}\, |=r=\sqrt{2} }
Hence , speed = \sqrt{2}\omega
