A shelf moves vertically with simple harmonic motion whose time period of motion is T seconds. Find the maximum amplitude that it can have so that objects resting on the shelf may always remain in contact with it.

Explanation

When the book is resting on the slab, its acceleration is same as that of the slab.
If m is the mass of the book and N is the normal reaction of the slab on the book, then according to Newton.s second law, the equation of motion of the book during upward motion of the slab is
`ma = N - mg = -momega^(2)y`
`rArr N = mg - momega^(2)y = m(g - omega^(2)y)`
Also, `omega = 2pi//T`
`therefore N = mg = (4pi^(2))/(T^(2)y)`
the book will not remain in touch with the lsab if N = 0 which gives
` g = (4pi^(2))/(T^(2))y`
`rArr y = (gT^(2))/(4pi^(2))`
For the amplitude grater than this, book will leave its contact with the slab.