Answer:
a) v = 0.79 m / s, a = 8.95 m / s², b) v = -0.3285 m / s
Step-by-step explanation:
A simple harmonic motion is described by the expression
x = A cos (wt + Ф)
the range of motion is A = 7.0 cm = 0.070 m
angular velocity and frequency are related
w = 2πf
w = 2π 1.80
w = 11.3 rad / s
we substitute
x = 0.070 cos (11.3t +Ф)
a) to find the velocity we use
v = dx / dt
v = - Aw sin (wt + Ф)
the maximum velocity when the cosine argument is π/2 0 3π/2 therefore the sine function is ±1
v = A w
v = 0.070 11.3
v = 0.79 m / s
the acceleration is
a = dv / dt
a = - A w² cos (wt + Ф)
the acceleration is maximum for an angle of o or pi, consequently the cosine works worth ±1
a = A w²
a = 0.070 11.3²
a = 8.95 m / s²
b) Let's find the time it takes to get to x = 0.03 m
wt + Ф = x / A
wt + Ф = 0.03 / 0.07 = 0.42857
To find the value of fi the initial conditions are used, in general if the system is released from rest fi = 0
t = 0.42857 / w
t = 0.42857 / 11.3
t = 0.0379 s
the speed for this time is
v = -A w sin wt
v = - 0.07 11.3 sin (11.3 0.0379)
remember angles are in radians
v = -0.3285 m / s