Question

A block is placed on a horizontal plank. The plank is performing $SHM$ along a vertical line with amplitude of $40\, cm$. The block just loses contact with the plank when the plank is momentarily at rest. Then :

• A. The period of its oscillations is $\frac{2 \pi}{5}$ sec
• B. The block weighs on the plank double its weight, when the plank is at one of the positions of momentary rest
• C. The block weighs 1.5 times its weight on the plank, halfway down from the mean position
• D. The block weighs is true weight on the plank, when velocity of the plank is maximum

Answer 1

Answer: (A), (B), (C), (D)

Block loses contact at the highest point. Then

$m g=m a\, \omega^{2} \,\,\,...(i)$
$\Rightarrow \omega=\sqrt{\frac{g}{a}}=\sqrt{\frac{10}{0.4}}=5\, rad\, s ^{-1} $
$\Rightarrow T=\frac{2 \pi}{5} s$
At lowest point
$N=m g+m a \omega^{2} $
$N=2 m g$ ( from(1))
Halfway down from mean position,
$N =m g+k\left(\frac{a}{2}\right) $
$=m g+\frac{m \omega^{2} a}{2} $
$=1.5 \,mg$
Block has maximum velocity when displacement (and thus acceleration) is zero, and thus has $N=m g$.