Answer:
v = 11.34 m / s
Explanation:
The simple harmonic motion of a spring and a mass is described by the equation
x = A cos (wt + Фfi)
where A is the amplitude of movement in this case 0.25 m, w the angular velocity and fi the initial phase
the angular velocity is
w = √ k / m
We can use Hooke's law to find the constant ka
F = k x
where the force is 300N and the spring stretch is 0.25m
k = F / x
k = 300 / 0.25
k = 1200 N / m
To find the phase angle di, let's use the system speed
va = dx /dt
va = A w sin (wt + Ф)
they tell us that the spring comes out of rest at time zero
Vd = Aw sin Ф
the only way this term is zero is that the angle Ф = 0
substitutions in the first equations
x = A cos wt
with
w = √RA (1200 / 0.5)
w = 48.99 rad / sec
we substitute in the first equations
x = 0.25 cos (48.99 t)
speed is
v = 0.25 48.99 without 48.99i
ask the speed for x = 0.15 m
we start by calculating the time it takes to get to this point
x = A cos wt
t = 1 / w cos-1 x / A
we look for the time
t = 1 / 48.99 cos-1 (0.15 / 0.25)
t = 0.0189 s
this is the first time it takes to get to the requested point
now we can calculate the speed
v = Aw sin (wt)
v = 0.25 48.99 sin (48.99 0.0189)
v = 11.34 m / s
