A spring and block are in the arrangement of Fig. 8-12. When the block is pulled out to `x= +4.0 cm`, we must apply a force of magnitude 360 N to hold it there. We pull the block to `x=11` cm and then release it. How much work does the spring do on the block as the block moves from `x_(i)= +5.0cm` to (a) `x= +3.0 cm`, (b) `x = 3.0 cm`, (c) `x= -5.0 cm`, and (d) `x = -9.0 cm` ?

Answer

To solve the problem, we need to calculate the work done by the spring on the block as it moves from an initial position \( x_i \) to a final position \( x_f \). The work done by the spring can be calculated using the formula:

\[ W = -\frac{1}{2} k (x_f^2 - x_i^2) \]

Where:
- \( W \) is the work done by the spring,
- \( k \) is the spring constant,
- \( x_i \) is the initial position,
- \( x_f \) is the final position.

### Step 1: Determine the Spring Constant \( k \)

From the information given, when the block is pulled to \( x = +4.0 \, \text{cm} \) and held there, a force of \( 360 \, \text{N} \) is required. The spring force is given by:

\[ F = kx \]

Where \( x = 0.04 \, \text{m} \) (since \( 4.0 \, \text{cm} = 0.04 \, \text{m} \)). Rearranging gives:

\[ k = \frac{F}{x} = \frac{360 \, \text{N}}{0.04 \, \text{m}} = 9000 \, \text{N/m} \]

### Step 2: Calculate Work Done for Each Case

#### (a) From \( x_i = +5.0 \, \text{cm} \) to \( x_f = +3.0 \, \text{cm} \)

Convert cm to m:
- \( x_i = 0.05 \, \text{m} \)
- \( x_f = 0.03 \, \text{m} \)

Now, substitute into the work formula:

\[ W = -\frac{1}{2} \cdot 9000 \cdot (0.03^2 - 0.05^2) \]

Calculating the squares:

\[ W = -\frac{1}{2} \cdot 9000 \cdot (0.0009 - 0.0025) = -\frac{1}{2} \cdot 9000 \cdot (-0.0016) = 7.2 \, \text{J} \]

#### (b) From \( x_i = +5.0 \, \text{cm} \) to \( x_f = -3.0 \, \text{cm} \)

Convert cm to m:
- \( x_f = -0.03 \, \text{m} \)

Now substitute into the work formula:

\[ W = -\frac{1}{2} \cdot 9000 \cdot ((-0.03)^2 - (0.05)^2) \]

Calculating:

\[ W = -\frac{1}{2} \cdot 9000 \cdot (0.0009 - 0.0025) = -\frac{1}{2} \cdot 9000 \cdot (-0.0016) = 7.2 \, \text{J} \]

#### (c) From \( x_i = +5.0 \, \text{cm} \) to \( x_f = -5.0 \, \text{cm} \)

Convert cm to m:
- \( x_f = -0.05 \, \text{m} \)

Now substitute into the work formula:

\[ W = -\frac{1}{2} \cdot 9000 \cdot ((-0.05)^2 - (0.05)^2) \]

Calculating:

\[ W = -\frac{1}{2} \cdot 9000 \cdot (0.0025 - 0.0025) = 0 \, \text{J} \]

#### (d) From \( x_i = +5.0 \, \text{cm} \) to \( x_f = -9.0 \, \text{cm} \)

Convert cm to m:
- \( x_f = -0.09 \, \text{m} \)

Now substitute into the work formula:

\[ W = -\frac{1}{2} \cdot 9000 \cdot ((-0.09)^2 - (0.05)^2) \]

Calculating:

\[ W = -\frac{1}{2} \cdot 9000 \cdot (0.0081 - 0.0025) = -\frac{1}{2} \cdot 9000 \cdot 0.0056 = -25.2 \, \text{J} \]

### Summary of Work Done
- (a) \( W = 7.2 \, \text{J} \)
- (b) \( W = 7.2 \, \text{J} \)
- (c) \( W = 0 \, \text{J} \)
- (d) \( W = -25.2 \, \text{J} \)