In the answer below, I have assumed the question refers to simple harmonic motion:
For simple harmonic motion, we have the formulas:
#omega=sqrt(k/m)#
and
#omega=2pif#
where #omega# is the angular velocity of the object, #k# is the spring constant, #m# is the mass of the object, and #f# is frequency
By combining the two equations and solving for #f#, we get:
#2pif=sqrt(k/m)#
#f=1/(2pi)sqrt(k/m)#
Since the only values we care about in this problem are #m# and #f#, we can disregard the constant #1/(2pi)# and let #k# be some arbitrary constant, say #1#, just to make this easier:
#f=sqrt(1/m)#
Now we can substitute #m# for #{m, 1/4m, 4m}#
If #m=m#:
#f=sqrt(1/m)#
this is our value to which we will compare quartering and quadrupling the mass to
If #m=1/4m#
#f=sqrt(1/(1/4m))#
#f=sqrt(4/m)#
#f=2sqrt(1/m)#
which is a frequency #2# times the original frequency.
if #m=4m#
#f=sqrt(1/(4m))#
#f=1/2sqrt(1/m)#
which is a frequency #1/2# times the original frequency.
Therefore, when we take one-fourth of the mass, we have a frequency #2# times the original frequency and when we take four times the mass, we have a frequency #1/2# times the original frequency.
