## A solid metal block with a mass of 1.30 kg is attached to a spring and is able to oscillate horizontally with negligible friction. The block

A solid metal block with a mass of 1.30 kg is attached to a spring and is able to oscillate horizontally with negligible friction. The block is pulled to a distance of 0.200 m from its equilibrium position, held in place with a force of 19.0 N, and then released from rest. It then oscillates in simple harmonic motion. (The block oscillates along the x-axis, where x = 0 (e) What is the maximum acceleration of the block? (Enter the magnitude in m/s2.) m/s2 (f) At what position(s) (in m) on the x-axis does the maximum acceleration occur? (g) What is the total mechanical energy of the oscillating spring-block system (inJ)? h) What is the speed of the block (in m/s) when its position is equal to one-third of the maximum displacement from equilibrium? m/s () What is the magnitude of the acceleration of the block (in m/s) when its position is equal to one-third of the maximum displacement from equilibrium? m/s2 is the equilibrium position.)

## Answers ( )

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