The spring in a retractable ballpoint pen is 2.0 cm long, with a 282 N/m spring constant. When the pen is retracted, the spring is compressed by 1.2 mm. When you click the button to extend the pen, you compress the spring by an additional 5.3 mm. How much energy is required to extend the pen? Write your answer out to 5 decimal places, using proper rounding.

To find the energy required to extend the pen, we need to calculate the change in the elastic potential energy stored in the spring.

The elastic potential energy \( U \) stored in a spring is given by:

\[
U = \frac{1}{2} k x^2
\]

where:
- \( k \) is the spring constant (282 N/m),
- \( x \) is the displacement from the spring's natural length.

### Step 1: Calculate the initial energy when the pen is retracted

The initial compression of the spring is \( x_1 = 1.2 \text{ mm} = 0.0012 \text{ m} \).

The initial potential energy \( U_1 \) is:

\[
U_1 = \frac{1}{2} \times 282 \times (0.0012)^2
\]

### Step 2: Calculate the energy when the pen is fully extended

The total compression after extending the pen is \( x_2 = 1.2 \text{ mm} + 5.3 \text{ mm} = 6.5 \text{ mm} = 0.0065 \text{ m} \).

The potential energy \( U_2 \) when the pen is fully extended is:

\[
U_2 = \frac{1}{2} \times 282 \times (0.0065)^2
\]

### Step 3: Calculate the energy required to extend the pen

The energy required to extend the pen is the difference between the final and initial potential energies:

\[
\Delta U = U_2 - U_1
\]

Let's calculate this step by step.

#### Initial Potential Energy \( U_1 \):

\[
U_1 = \frac{1}{2} \times 282 \times (0.0012)^2 = 0.00020304 \text{ J}
\]

#### Final Potential Energy \( U_2 \):

\[
U_2 = \frac{1}{2} \times 282 \times (0.0065)^2 = 0.00595935 \text{ J}
\]

#### Energy Required \( \Delta U \):

\[
\Delta U = 0.00595935 \text{ J} - 0.00020304 \text{ J} = 0.00575631 \text{ J}
\]

### Final Answer

The energy required to extend the pen is approximately **0.00576 J** to 5 decimal places (after proper rounding).

Would you like more details or have any questions? Here are 5 questions that might help you expand your understanding:

1. What is the physical significance of the spring constant in this problem?
2. How would the energy change if the spring constant were doubled?
3. What would happen to the energy if the initial compression were less?
4. How does the elastic potential energy depend on the displacement?
5. Can you derive the formula for elastic potential energy starting from Hooke's law?

**Tip:** When dealing with small displacements and energy calculations, always keep track of unit conversions to avoid errors.

Learn how to calculate the energy required to extend a retractable pen's spring. This problem involves using Hooke's law and the formula for elastic potential energy. Suitable for high school physics students.

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