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 objects are resting on the shelf, then the accelerations of th eobjects will be the same as that of the shelf.
If R is the reaction of the shelf on the object of mass m placed on the shelf , then the equation of motion of a mass ono the shelf during upward motion of the shelf is
`R-mg=massxxac cel eration=-momega^(2)y`
or `R=mg-momega^(2)y=m(g-omega^(2)y)`
`=m(g-(4pi^(2))/(T^(2))y)`
The object will just leave the shelf if `R=0`,
i.e., `g=(4pi^(2))/(T^(2))y` or `y=(gT^(2))/(4pi^(2))`
This is the gratest amplitude the shelf should have during its vibration. For amplitude greater than `(gT^(2))/(4pi^(2))` , the object will leave the shelf.