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To solve this problem, let's go through it step-by-step by analyzing the forces involved and applying energy conservation principles. We have to find the coefficient of friction, \( \mu \), on the incline.

### Given Data
1. **Mass of the box** (\( m \)): \( 0.450 \, \text{kg} \)
2. **Spring compression length** (\( x \)): \( 0.198 \, \text{m} \)
3. **Spring constant** (\( k \)): \( 1350 \, \text{N/m} \)
4. **Incline length** (\( d \)): \( 1.65 \, \text{m} \)
5. **Angle of incline** (\( \theta \)): \( 42.5^\circ \)
6. **Maximum height reached** (\( H \)): \( 3.25 \, \text{m} \)

### Approach to Solution

1. **Step 1**: Calculate the initial energy provided by the spring.
   \[
   E_{\text{spring}} = \frac{1}{2} k x^2
   \]
   
2. **Step 2**: Determine the energy required to reach the maximum height \( H \). When the box reaches the maximum height, all the energy (after overcoming friction) is converted into gravitational potential energy at height \( H \).
   \[
   E_{\text{gravity}} = m g H
   \]
   
3. **Step 3**: Calculate the work done by friction along the incline. The work done by friction (\( W_{\text{friction}} \)) is given by:
   \[
   W_{\text{friction}} = f_k \cdot d = \mu \cdot N \cdot d
   \]
   where \( N = m g \cos \theta \) is the normal force.

4. **Step 4**: Apply energy conservation, equating the initial energy from the spring to the work done against friction and the gravitational potential energy:
   \[
   E_{\text{spring}} = E_{\text{gravity}} + W_{\text{friction}}
   \]
   Substituting the expressions:
   \[
   \frac{1}{2} k x^2 = m g H + \mu m g \cos \theta \cdot d
   \]
   
5. **Step 5**: Solve for \( \mu \), the coefficient of friction.

Let's go through the calculations explicitly.

#### Step-by-Step Calculation

1. **Calculate the spring energy**:
   \[
   E_{\text{spring}} = \frac{1}{2} \cdot 1350 \cdot (0.198)^2
   \]
   \[
   E_{\text{spring}} \approx 26.5053 \, \text{J}
   \]

2. **Calculate the gravitational potential energy at height \( H \)**:
   \[
   E_{\text{gravity}} = 0.450 \cdot 9.81 \cdot 3.25
   \]
   \[
   E_{\text{gravity}} \approx 14.3318 \, \text{J}
   \]

3. **Substitute values into the energy equation to solve for \( \mu \)**:
   \[
   26.5053 = 14.3318 + \mu \cdot 0.450 \cdot 9.81 \cdot \cos(42.5^\circ) \cdot 1.65
   \]

Solving this equation will give the value of \( \mu \). Let me calculate it.The coefficient of friction (\( \mu \)) on the inclined surface is approximately \( 2.26 \).

Would you like further details or have any questions about this solution?

Here are five related questions that can help deepen understanding:

1. How would the result change if the angle of the incline was steeper?
2. What happens to the coefficient of friction if the spring constant \( k \) is increased?
3. How would we solve this problem if the incline was frictionless?
4. What role does the gravitational potential energy play in determining the maximum height \( H \)?
5. Could this problem be solved using kinematic equations instead of energy conservation?

**Tip**: Remember, frictional force depends on both the coefficient of friction and the normal force, which is affected by the incline angle.

A resting box of mass m = 0.450 kg is launched from a spring with a compression length x = 0.198 m and force constant k = 1350 N/m. The box slides on a rough inclined plane of length d = 1.65 m at an angle of θ = 42.5°. Upon reaching the top, it acts as a projectile, reaching a maximum height H = 3.25 m from the ground. What is the coefficient of friction of the inclined surface?

This problem involves calculating the coefficient of friction on an inclined plane using energy conservation principles. Given data includes mass (0.450 kg), spring constant (1350 N/m), incline length (1.65 m), angle (42.5°), and maximum height (3.25 m). Follow the solution to understand spring potential energy, gravitational potential energy, and work done by friction.

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