In considering question #4, how fast is the mass moving when it is half way between the equilibrium and maximum amplitude, AND what is the kinetic and potential energies at that location?

Question

Here is Question Four:

4. A 27.8 gram mass oscillates on the end of a spring and its motion can be described by: x = (1.5) * sin (21.4t) [here the units are standard, meters and seconds] What are the following for this system: amplitude, frequency, period, spring constant, max velocity and total energy?

Answer 1

You may have already answered question 4 itself, but here goes as a check (I need some of this anyway to have a value for spring constant k):  Given amplitude A = 1.5 m, mass = 0.0278 kg, ω = 2πf = 21.4 rad/s (coefficient of t in sin expression), then f = 3.406 Hz, T = 1/f = 0.2936 s. For spring SHM, ω = √(k/m) so k = mω² = 12.73 N/m.

Max velocity = Aω = 32.1 m/s. Total energy E_t = ½ m v_max² = ½ k A² = 14.32 J.

Halfway to maximum amplitude, spring potential energy E_s = ½ k (½ A)² = 3.58 J, and KE = the remaining 10.74 J. The speed at this point comes from  v² = 2 KE/m = 773, v = 27.8 m/s.