Trigonometric Derivative Interpretation! Need Help?

Question

The function y = A sin((square root of (k/m) * t ) represents the oscilattion of a mass m at the end of a spring. The constant k measures the stiffness of the spring.
a. Find a time at which the mass is farthest from it's equilibrium
position. Find a time at which the mass is moving fastest. Find
a time at which the mass is accelerating fastest.
b. What os the period, T, of the oscillation?
c. Find dT/dm. What does the sign of dT/dm tell you?

Answer 1

a.
d(sin(at))/dt = a cos at, a=(k/m)^0.5
let a cos at = 0
at=arccos 0, t= (arccos(0))/a = (arccos(0))*(m/k)^0.5

every time cos at goes through 0, the mass is as far as it get s from equilibrium. This will also be the time of greatest acceleration.

b.
T==1/f = 1/(k/m)^0.5 = (m/k)^0.5

c.
dT/dm = (0.5(m/k)^-0.5)(1/k), Period increases as m^0.5 increases