ω = sqrt( k/m )
ω: angular frequency; m: mass; k: spring constant
ω = 2π/T
T: period (how long it takes for one revolution)
T = 4.59s/10rev = 0.459s/rev
ω = 2π/T = ω = 2π/0.459
ω = 13.6889 radians/s
One revolution, or 2π radians, is 8 cm. (4 cm to stretch out, the same 4 cm to retract)
13.6889 radians/s * (4cm/2πradians) = 86.01 cm/s
ω = 86.01 cm/s = 0.086 m/s
Let's take the first example, where mass is 50g and angular freq. is 4.59s.
50g = 0.05 kg (kilograms are standard in SI, not grams, and are used more directly in calculations relating to force)
ω = sqrt( k/m )
0.086 = sqrt( k/0.05 )
Square both sides
0.007396 = k/0.05
multiply both by 0.05
k = 0.0003698 N/m = 3.698 * 10^-4 N/m
Let's graph frequency in relation to mass (mass is the independent variable)
Let's calculate the slope of the period-to-mass graph
m (slope) = (T2 - T1)/(m2 - m1) (change in frequency over change in mass)
Let's use the second example for T2 and T2, and the first for ω1 and m1.
m = (6.32-4.59)/(100-50) = 1.73/50
m = 0.0346 seconds/gram, which means for every gram of mass added, the period for 10 revolutions increases by 0.0346 seconds.
