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To solve this problem, we need to apply the principles of energy conservation and the equations of projectile motion. Here’s the breakdown of the solution:

### Given Data
1. Mass of the box, \( m = 0.450 \, \text{kg} \)
2. Spring compression, \( x = 0.198 \, \text{m} \)
3. Spring constant, \( k = 1350 \, \text{N/m} \)
4. Distance along the incline, \( d = 1.65 \, \text{m} \)
5. Angle of incline, \( \theta = 42.5^\circ \)
6. Maximum height of the projectile, \( H = 3.25 \, \text{m} \)
7. Coefficient of friction on the incline, \( \mu \) (to be determined)

### Solution Steps

1. **Energy Stored in the Spring:**
   The initial energy in the spring is given by:
   \[
   E_{\text{spring}} = \frac{1}{2}kx^2
   \]
   Substituting the values:
   \[
   E_{\text{spring}} = \frac{1}{2} \times 1350 \times (0.198)^2
   \]
   Calculate \( E_{\text{spring}} \).

2. **Energy at the Top of the Incline (before the box leaves the ramp):**
   By the work-energy theorem, the initial energy stored in the spring is used to overcome:
   - The gravitational potential energy gain along the incline
   - The work done against friction

   Let \( E_{\text{top}} \) be the kinetic energy at the top of the incline. We have:
   \[
   E_{\text{spring}} = m g d \sin(\theta) + \mu m g d \cos(\theta) + E_{\text{top}}
   \]

   Rearranging to solve for \( E_{\text{top}} \):
   \[
   E_{\text{top}} = E_{\text{spring}} - m g d \sin(\theta) - \mu m g d \cos(\theta)
   \]

3. **Projectile Motion Analysis:**
   At the top of the incline, the box transitions into projectile motion. The vertical height \( H \) reached is given by the initial vertical velocity component squared divided by \( 2g \):
   \[
   H = \frac{(v_y)^2}{2g}
   \]
   where \( v_y = v \sin(\theta) \) and \( v \) is the speed at the top of the incline. 

4. **Calculating \( \mu \):**
   By substituting the values for \( H \), solving for \( v_y \) and using energy equations, we can find

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

Solve a physics problem involving energy conservation and friction. Given a box launched by a spring with known compression and spring constant, this problem explores the coefficient of friction as it slides up an inclined plane before becoming a projectile. Includes analysis with the work-energy theorem and projectile motion.

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