Question

A block is placed on a horizontal plank. The plank is performing $SHM$ along a vertical line with amplitude of $40cm$. 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

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