A small mass is attached to a vertical spring as shown. The mass is displaced downward a small distance from the equilibrium point and released from rest. Set the origin of the coordinate system at the equilibrium position of the object and choose upward as the positive direction. Assume air resistance is so small that it can be ignored.

Assume that the mass $($ m $)$ is oscillating with the period, $T=4s.$

$($a$)$ What is the period if the mass is halved? $($Note: You do not know the values of either k or m $.$ Do not assume any particular values for them. The required analysis involves thinking about ratios$)$

$($b$)$ What is the period if the value of the spring constant is quadrupled?

$($c$)$ What is the period if the oscillation amplitude is doubled while m and k are unchanged?