2. To stretch a spring 3.00 cm from its unstretched length, 12.0 J of

work must be done. (a) What is the force constant of this spring? (b)

What magnitude force is needed to stretch the spring 3.00 cm from its

unstretched length? (c) How much work must be done to compress this

spring 4.00 cm from its unstretched length, and what force is needed to

compress it this distance?

### Given:
- Stretch distance, \( x = 3.00 \, \text{cm} = 0.0300 \, \text{m} \)
- Work done to stretch, \( W = 12.0 \, \text{J} \)

#### (a) Force constant of the spring

The work done to stretch a spring is related to the spring constant \( k \) and the displacement \( x \) by the formula:

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

We can rearrange this equation to solve for \( k \):

\[
k = \frac{2W}{x^2}
\]

Substitute the values:

\[
k = \frac{2 \times 12.0 \, \text{J}}{(0.0300 \, \text{m})^2}
\]

\[
k = \frac{24.0 \, \text{J}}{0.0009 \, \text{m}^2}
\]

\[
k = 26,666.67 \, \text{N/m}
\]

So, the force constant \( k \) is approximately \( 26,667 \, \text{N/m} \).

---

#### (b) Force needed to stretch the spring 3.00 cm

The force \( F \) needed to stretch or compress a spring is given by Hooke's law:

\[
F = kx
\]

Substituting the values of \( k \) and \( x \):

\[
F = 26,667 \, \text{N/m} \times 0.0300 \, \text{m}
\]

\[
F = 800.0 \, \text{N}
\]

So, the force needed to stretch the spring 3.00 cm is \( 800.0 \, \text{N} \).

---

#### (c) Work and force needed to compress the spring 4.00 cm

1. **Work done to compress the spring 4.00 cm:**

We use the same work formula, but with \( x = 4.00 \, \text{cm} = 0.0400 \, \text{m} \):

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

Substitute the values:

\[
W = \frac{1}{2} \times 26,667 \, \text{N/m} \times (0.0400 \, \text{m})^2
\]

\[
W = 0.5 \times 26,667 \times 0.0016 \, \text{m}^2
\]

\[
W = 21.33 \, \text{J}
\]

So, the work required to compress the spring 4.00 cm is \( 21.33 \, \text{J} \).

2. **Force needed to compress the spring 4.00 cm:**

We use Hooke’s law again:

\[
F = kx
\]

Substitute the values:

\[
F = 26,667 \, \text{N/m} \times 0.0400 \, \text{m}
\]

\[
F = 1,066.68 \, \text{N}
\]

So, the force needed to compress the spring 4.00 cm is approximately \( 1,067 \, \text{N} \).

---

### Summary:
- (a) Force constant \( k = 26,667 \, \text{N/m} \)
- (b) Force to stretch 3.00 cm: \( 800.0 \, \text{N} \)
- (c) Work to compress 4.00 cm: \( 21.33 \, \text{J} \), Force to compress 4.00 cm: \( 1,067 \, \text{N} \)

Would you like more details or have any questions?

### Related Questions:
1. How does the work done change if the spring is stretched by 6.00 cm instead of 3.00 cm?
2. What would the force constant be if the work done to stretch the spring by 3.00 cm were doubled?
3. How does the energy stored in a spring change when it is compressed instead of stretched?
4. If a different spring has a force constant of 50,000 N/m, what force would be required to stretch it by 2.00 cm?
5. How does the Hooke’s law equation change when considering a spring system in parallel?

**Tip**: For springs, the work done is proportional to the square of the displacement, meaning small changes in stretch or compression result in significant changes in energy stored.

To stretch a spring 3.00 cm from its unstretched length, 12.0 J of work must be done. (a) What is the force constant of this spring? (b) What magnitude force is needed to stretch the spring 3.00 cm from its unstretched length? (c) How much work must be done to compress this spring 4.00 cm from its unstretched length, and what force is needed to compress it this distance?

Learn how to calculate the force constant of a spring, the force needed to stretch or compress it, and the work done in the process. This detailed solution uses Hooke's law and work-energy theorem, ideal for high school physics students. Example: stretching a spring 3.00 cm with 12.0 J of work.

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