The frequency (f) of an oscillating spring is related to the mass (m) of the object attached to the spring and the spring constant (k) by the following equation:
f = (1/2π) * sqrt(k/m)
where π is pi (approximately equal to 3.14159).
To find the mass required for a spring with a spring constant of 2.000 x 10^3 N/m to oscillate 5.0 times per second, we can rearrange this equation to solve for m:
m = k / (4π^2 * f^2)
Substituting in the given values, we get:
m = (2.000 x 10^3 N/m) / (4π^2 * (5.0/s)^2) = 0.0255 kg
Therefore, the mass required for the spring to oscillate 5.0 times per second is 0.0255 kg (or approximately 25.5 grams).
Similarly, to find the mass required for the spring to oscillate 10.0 times per second, we can use the same equation:
m = (2.000 x 10^3 N/m) / (4π^2 * (10.0/s)^2) = 0.00638 kg
Therefore, the mass required for the spring to oscillate 10.0 times per second is 0.00638 kg (or approximately 6.38 grams).