To calculate the initial potential energy stored in the spring ($U_{\text{spring}}$), we consider the energy transformations that occur. The spring's potential energy is used to:

1. Provide the block with kinetic energy at point B.
2. Work against the force of friction as the block moves up the incline.
3. Increase the block's gravitational potential energy as it rises along the incline.

Step 1: Kinetic Energy at Point B

The kinetic energy ($KE$) at point B is given by: $KE = \frac{1}{2} m v^2$ Substituting $m = 1.50 \, \text{kg}$ and $v = 7.00 \, \text{m/s}$: $KE = \frac{1}{2} (1.50) (7.00)^2 = 36.75 \, \text{J}$

Step 2: Work Done Against Friction

The work done against friction ($W_{\text{friction}}$) is: $W_{\text{friction}} = f_k d = \mu_k N d$ The normal force $N$ on an incline is: $N = m g \cos \theta$ Substituting $m = 1.50 \, \text{kg}$, $g = 9.8 \, \text{m/s}^2$, and $\theta = 30^\circ$: $N = (1.50)(9.8)(\cos 30^\circ) = 12.73 \, \text{N}$ Thus, the frictional force is: $f_k = \mu_k N = (0.50)(12.73) = 6.365 \, \text{N}$ The work done against friction over a distance of $d = 6.00 \, \text{m}$ is: $W_{\text{friction}} = (6.365)(6.00) = 38.19 \, \text{J}$

Step 3: Gravitational Potential Energy

The increase in gravitational potential energy ($\Delta U_{\text{gravity}}$) is: $\Delta U_{\text{gravity}} = m g h$ The height $h$ along the incline is related to the distance $d$ by: $h = d \sin \theta$ Substituting $d = 6.00 \, \text{m}$ and $\theta = 30^\circ$: $h = 6.00 \sin 30^\circ = 3.00 \, \text{m}$ Thus: $\Delta U_{\text{gravity}} = (1.50)(9.8)(3.00) = 44.10 \, \text{J}$

Step 4: Total Energy from the Spring

The total initial energy stored in the spring is the sum of the above: $U_{\text{spring}} = KE + W_{\text{friction}} + \Delta U_{\text{gravity}}$ Substituting the values: $U_{\text{spring}} = 36.75 + 38.19 + 44.10 = 119.04 \, \text{J}$

Final Answer:

The amount of potential energy initially stored in the spring is: $\boxed{119.0 \, \text{J}}$

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Would you like a breakdown of the concepts or further help? 😊
Here are some related questions to explore:

1. How would the result change if the coefficient of kinetic friction were 0.30 instead of 0.50?
2. What is the maximum height the block could reach if the spring energy were doubled?
3. How would a steeper incline (e.g., 45°) affect the total work against friction?
4. Can we calculate the spring constant if the spring's compression is known?
5. What happens if the block were sliding down instead of up the incline?

Tip: Always break problems into small parts (e.g., energy, forces) for easier analysis!