A force of 10 N holds an ideal spring with a `20 N/m` spring constant in compression. The potential energy stored in the spring is

Options

A. `0.5`J
B. `2.5`J
C. `5`J
D. 10 J

Answer

To find the potential energy stored in the spring when a force is applied, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Force (F) = 10 N
- Spring constant (k) = 20 N/m

2. **Use Hooke's Law:**
Hooke's Law states that the force exerted by a spring is proportional to its displacement (x) and is given by the formula:
\[ F = kx \]
Here, we can rearrange this formula to find the displacement (x):
\[ x = \frac{F}{k} \]

3. **Calculate the Displacement (x):**
Substitute the values of F and k into the equation:
\[ x = \frac{10 \, \text{N}}{20 \, \text{N/m}} = 0.5 \, \text{m} \]

4. **Calculate the Potential Energy (PE):**
The potential energy stored in a compressed or stretched spring is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]
Substitute the values of k and x into the equation:
\[ PE = \frac{1}{2} \times 20 \, \text{N/m} \times (0.5 \, \text{m})^2 \]

5. **Simplify the Calculation:**
Calculate \( (0.5)^2 \):
\[ (0.5)^2 = 0.25 \]
Now substitute this value back into the equation for potential energy:
\[ PE = \frac{1}{2} \times 20 \times 0.25 \]
\[ PE = \frac{1}{2} \times 5 = 2.5 \, \text{J} \]

6. **Conclusion:**
The potential energy stored in the spring is:
\[ PE = 2.5 \, \text{J} \]