Certainly! Let's find the potential energy stored in the spring. We will use the formula for the elastic potential energy stored in a compressed or stretched spring, which is given by:
[tex]\[ PE = \frac{1}{2} k x^2 \][/tex]
where:
- [tex]\( PE \)[/tex] is the potential energy stored in the spring,
- [tex]\( k \)[/tex] is the spring constant (in N/m),
- [tex]\( x \)[/tex] is the displacement from the equilibrium position (in meters).
Here are the steps to solve the problem:
### 1. Identify the given values
- The spring constant [tex]\( k \)[/tex] is given as 115 N/m.
- The displacement [tex]\( x \)[/tex] is given as 1.9 cm.
### 2. Convert displacement to meters
Since the spring constant is in N/m, we need the displacement [tex]\( x \)[/tex] in meters. Convert 1.9 cm to meters:
[tex]\[ 1.9 \, \text{cm} = \frac{1.9}{100} \, \text{m} = 0.019 \, \text{m} \][/tex]
### 3. Substitute the values into the formula
Now, use the values [tex]\( k = 115 \, \text{N/m} \)[/tex] and [tex]\( x = 0.019 \, \text{m} \)[/tex] in the formula:
[tex]\[ PE = \frac{1}{2} k x^2 \][/tex]
[tex]\[ PE = \frac{1}{2} \times 115 \, \text{N/m} \times (0.019 \, \text{m})^2 \][/tex]
### 4. Calculate the squared displacement
First, calculate the value of [tex]\( (0.019 \, \text{m})^2 \)[/tex]:
[tex]\[ (0.019 \, \text{m})^2 = 0.019 \times 0.019 = 0.000361 \, \text{m}^2 \][/tex]
### 5. Compute the potential energy
Now, multiply this value by the spring constant and then by 0.5:
[tex]\[ PE = \frac{1}{2} \times 115 \, \text{N/m} \times 0.000361 \, \text{m}^2 \][/tex]
### 6. Simplify the expression
[tex]\[ PE = 0.5 \times 115 \times 0.000361 \][/tex]
[tex]\[ PE = 57.5 \times 0.000361 \][/tex]
[tex]\[ PE = 0.0207575 \, \text{J} \][/tex]
### 7. Round the result
For a final answer, we can round this to a suitable number of significant figures. Given the precision of the input values, we'll consider two decimal places:
[tex]\[ PE \approx 0.021 \, \text{J} \][/tex]
Therefore, the potential energy stored in the spring is approximately [tex]\( 0.021 \)[/tex] joules.
