To solve the problem step by step, we will analyze the forces acting on the mass when it is in simple harmonic motion.
### Step 1: Identify the given data
- Natural length of the spring, \( L_0 = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Force constant of the spring, \( k = 2.0 \times 10^3 \, \text{N/m} \)
- Mass of the body, \( m = 10 \, \text{kg} \)
- Gravitational acceleration, \( g = 10 \, \text{m/s}^2 \)
- Stretched length of the spring when pulled down, \( L = 58 \, \text{cm} = 0.58 \, \text{m} \)
### Step 2: Calculate the equilibrium position
At equilibrium, the force exerted by the spring equals the weight of the mass:
\[
mg = kx_0
\]
Where \( x_0 \) is the elongation from the natural length at equilibrium.
Calculating \( mg \):
\[
mg = 10 \, \text{kg} \times 10 \, \text{m/s}^2 = 100 \, \text{N}
\]
Now, substituting into the equilibrium equation:
\[
100 = 2000 \times x_0
\]
Solving for \( x_0 \):
\[
x_0 = \frac{100}{2000} = 0.05 \, \text{m} = 5 \, \text{cm}
\]
### Step 3: Determine the total elongation when pulled down
When the body is pulled down to a length of 58 cm, the total elongation \( x \) from the natural length is:
\[
x = L - L_0 = 0.58 \, \text{m} - 0.5 \, \text{m} = 0.08 \, \text{m} = 8 \, \text{cm}
\]
### Step 4: Calculate the net force at the lowermost position
At the lowermost position, the net force \( F_{\text{net}} \) acting on the mass is given by the difference between the weight of the mass and the spring force:
\[
F_{\text{net}} = mg - kx
\]
Substituting the values:
\[
F_{\text{net}} = 100 \, \text{N} - (2000 \, \text{N/m} \times 0.08 \, \text{m})
\]
Calculating the spring force:
\[
kx = 2000 \times 0.08 = 160 \, \text{N}
\]
Now substituting back:
\[
F_{\text{net}} = 100 \, \text{N} - 160 \, \text{N} = -60 \, \text{N}
\]
### Step 5: Relate the net force to the value of \( x \)
The problem states that the net force is \( 10x \) Newtons. Thus:
\[
-60 = 10x
\]
Solving for \( x \):
\[
x = -6 \, \text{N}
\]
Since we are interested in the magnitude:
\[
|x| = 6
\]
### Final Answer
The value of \( x \) is \( 6 \).
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