Respuesta :
Answer:
a) a = - 1.76 m / s², b) F = 20.5 N, c) w = 3.55 rad / s
Explanation:
a) a simple harmonic motion is described by the expression
x = A cos (wt + Ф)
in this case they give us the frequency
w = 2π f
w = 2π 0.40
w = 2.51 rad / s
as the maximum elongation is 0.65 m this corresponds to the amplitude of the movement
A = 0.65 m
to find the phase constant (Ф) we use the initial condition that for t = 0 v = 0 and x = A, we substitute
A = A cos (0+ Ф)
cos Ф = 1
Ф = 0
the resulting equation is
x = 0.65 cos (2.51 t)
Let's find the time it takes to get to x = 0.28 m
0.28 = 0.65 cos 2.51 t
2.51 t = cos-1 (0.28 / 0.65)
remember angles are in radians
t = 1.1254 / 2.51
t = 0.448 s
the acceleration is
[tex]a = \frac{dv}{dt} = \frac{d^2x}{dt^2}[/tex]
a = -A w² cos wt
we subtitle
a = - 0.65 2.51² cos (2.51 0.448)
a = - 1.76 m / s²
b) the maximum acceleration occurs when the cosine is ±1
a = A w²
a = 0.65 2.51²
a = 4.10 m / s²
Let's use Newton's second law
F = m a
F = 5 4.1
F = 20.5 N
c) The angular velocity is given by
w² = k / m
let's find the spring constant
k = m w²
k = 5 2.51²
k = 31.5 N / m
therefore if the block is exchanged for another with mass m'= 2.5 kg
w = √(31.5 / 2.5)
w = 3.55 rad / s
