Respuesta :
Answer:
e) a = 14.62 m / s², f) the maximum acceleration occurs at the points x = ± A , g) Em = 1.9 J , h) v = 1.61 m / s, i) a = -4.82 m / s²
Explanation:
A system of a spring with a mass is described by the equation
x = A cos (wt + Ф)
Where
w = √ k / m
e) ask to find the acceleration, for this we use the definition of acceleration
v = dx / dt
a = dv / dt
v = -A w cos (wt + Ф)
a = -A w² cos (wt + Ф)
The acceleration is maximum when the cosine is ± 1
a = A w²
In the exercise, indicate that the amplitude is A = 0.200 m
To find the constant k, let's use Newton's equilibrium equation
F - Fe = 0
F = k x
k = F / x
k = 19 / 0.2
k = 95 N / m
Now we can calculate the angular velocity
w = √ k / m
w =√ (95 / 1.3)
w = 8.55 rad / s
Substitutes in the maximum acceleration equation
a = 0.2 8.55²
a = 14.62 m / s²
f) the acceleration depends on the cosine and the cosine is maximum in the express of the displacement, therefore the maximum acceleration occurs at the points x = ± A
g) mechanical energy is given by the equation
Em = ½ k A²
Em = ½ 95 0.2²
Em = 1.9 J
h) what is the speed when the position is x = 1/3 A
Em = K + U
Em = ½ m v² + ½ k x²
v² = (Em - ½ k x²) 2 / m
v² = (1.9 - ½ 95 (0.2 /3)²) 2 / 1.30
v² = 2,598
v = 1.61 m / s
i) block acceleration when x = A / 3
For this we must find fi and the time to reach this position
Let's look fi
As the system starts from rest the speed starts is zero
V = A w sin (0+ Ф)
0 = sin Ф
Ф = 0
Now we look for the time to reach the displacement x = A / 3
A / 3 = A cos 8.5 t
1/3 = cos 8.5t
8.5t = cos-1 1/3
t = 1 / 8.5 cos-1 1/3
t = 0.1448 s
Now we substitute in the acceleration equation
a = - A w² cos wt
a = - 0.2 8.5² cos (8.5 0.1448)
a = -4.82 m / s²
