Given x(t) = xm cos 2ðt/T for a spring with constant k and attached mass m sketch as a function of t, (a) the displacement x, (b) the velocity v, (c) the acceleration.

Answer 1

x = A cos 2pit/T
dx/dt = -A (2pi/T)sin 2pit/T
d^2x/dt^2 = -A(2pi/T)^2 cos 2pit/T
=-(2pi/T)^2 x

F = -kx = m d^2x/dt^2
- k x = -m(2pi/T)2 x
so
2pi/T = sqrt(k/m) = w to pick a letter
then
x = A cos wt
dx/dt = velocity = -A w sin wt
d^2x/dt^2 = -w^2x which is -Aw^2cos wt