To calculate the coefficient of friction ($\mu$) given the provided parameters, we will use the principle that the loss in total mechanical energy is equal to the work done by friction over the distance traveled. Here's the detailed step-by-step calculation:

Given Data:

1. Angle of inclination: $\theta = 10.95^\circ$
2. Gravitational acceleration: $g = 9.81 \, \text{m/s}^2$
3. Mass of the cart: $m = 0.582 \, \text{kg}$
4. Distance difference: $d = 0.748 - 0.221 = 0.527 \, \text{m}$

Mechanical Energy Loss:

Assuming the loss in total mechanical energy ($\Delta E$) is due to friction, we compute:

Frictional Force:

The work done by friction over the distance $d$ is: $W_{\text{friction}} = \mu m g \cos\theta \cdot d$

Setting this equal to the loss of mechanical energy: $\Delta E = \mu m g \cos\theta \cdot d$

Solving for $\mu$: $\mu = \frac{\Delta E}{m g \cos\theta \cdot d}$

You need the value of $\Delta E$ (loss of mechanical energy) to proceed with the calculation. If you provide this value, I can calculate $\mu$ for you.

Would you like to share the value of $\Delta E$, or should I help estimate it based on further details?