A block attached to a spring is displaced from equilibrium to the position x=+2.8m and released. The period is 1.6s . At what positions and times during the first complete cycle do the following conditions occur.

Question

a) |v|=0.5vmax, Where vmax is the maximum speed?

First position:
First time:


Second position:
Second time:


Third position:
Third time:


Fourth position:
Fourth time:
s.

b) |a|=0.5amax , Where amax is the maximum magnitude of the acceleration?

First position:
First time:


Second position:
Second time:


Third position:
Third time:


Fourth position:
Fourth time:

Please helpp im desperate!!

Answer 1

cosine 0 =1 so
x = A cos 2 pi t/T = 0.28 cos 2 pi t/T
but T = 1.6
sso
x = 0.28 cos 3.93 t
note ---- 3.93 t is in radians not degrees
now
dx/dt = v = - 0.28 * 3.93 sin 3.93 t = -1.10 sin 3.93 t
and
a = d/dt v = -1.10* 3.93 * cos 3.93 t = -4.32 cos 3.93 t
==================== now plug and chug
for example |v| is max first when 3.93 t = pi/2 (or 90 degrees|
t = 3.14159 / (3.93*2) = 0.40 seconds
and velocity v is - 1.10 there)

Answer 2

if you want to know for exmple when v = 1/2 v max it its when the sin is 1/2
that is when the angle is 30 degrees which is pi/6 radians