A small block of mass m is placed on a bigger block of mass M that is attached to a vertical spring of spring constant k . The system oscillates vertically. Determine the maximum amplitude with which the two blocks can oscillate together without losing contact.

Step-by-step solution

Let the amplitude of oscillation be A . The angular frequency of the spring-mass system is given by = k M+m . The maximum downward acceleration of the system occurs at the highest point of oscillation and its magnitude is a_ = ^2 A = k M+m A . Consider the forces acting on the small block of mass m at the highest point: the downward gravitational force mg and the upward normal reaction N from the bigger block. The equation of motion for the small block is: mg - N = m a_ For the block of mass m to not lose contact w