(1) The position around equilibrium of an object in simple harmonic motion is described by
where
A is the amplitude of the motion
is the angular frequency.
The velocity is the derivative of the position:
where
is the maximum velocity of the object.
The acceleration is the derivative of the velocity:
where
is the maximum acceleration of the object.
We know from the problem both maximum velocity and maximum acceleration:
From the first equation, we get
(1)
and if we substitute this into the second equation, we find the angular fequency
while the amplitude is (using (1)):
(b) We found in the previous step that the angular frequency of the motion is
But the angular frequency is related to the period by
and so, the period is
where
A is the amplitude of the motion
The velocity is the derivative of the position:
where
The acceleration is the derivative of the velocity:
where
We know from the problem both maximum velocity and maximum acceleration:
From the first equation, we get
and if we substitute this into the second equation, we find the angular fequency
while the amplitude is (using (1)):
(b) We found in the previous step that the angular frequency of the motion is
But the angular frequency is related to the period by
and so, the period is