To find the friction coefficient of the horizontal surface, we can use the principles of energy conservation and frictional work.

Given Data

• Mass of the box, $m = 8.0 \, \text{kg}$
• Initial height, $y_0 = 0.25 \, \text{m}$
• Distance slid on rough surface, $d = 2.0 \, \text{m}$
• Gravitational acceleration, $g = 9.8 \, \text{m/s}^2$

Steps to Solve

1. Calculate the Initial Potential Energy: The box starts at height $y_0$ on a frictionless ramp, so it has gravitational potential energy given by: $PE = m \cdot g \cdot y_0$

2. Convert to Kinetic Energy at Bottom of Ramp: As the box reaches the bottom of the ramp, all its potential energy will convert into kinetic energy since there’s no friction on the ramp: $KE = m \cdot g \cdot y_0$

3. Work Done by Friction: When the box slides onto the rough horizontal surface, friction does work to stop the box. The work done by friction $W_f$ is: $W_f = \text{friction force} \times d = \mu \cdot m \cdot g \cdot d$ where $\mu$ is the friction coefficient we want to find.

4. Set up Energy Conservation Equation: Since the box stops after sliding $d = 2.0 \, \text{m}$, the initial kinetic energy at the bottom of the ramp equals the work done by friction: $m \cdot g \cdot y_0 = \mu \cdot m \cdot g \cdot d$

5. Solve for $\mu$: $\mu = \frac{y_0}{d}$ Substitute $y_0 = 0.25 \, \text{m}$ and $d = 2.0 \, \text{m}$: $\mu = \frac{0.25}{2.0} = 0.125$

Answer

The friction coefficient of the horizontal surface, $\mu$, is $0.125$.

Would you like a deeper explanation of each step?

Here are some related questions for further understanding:

1. How would the friction coefficient change if the initial height were doubled?
2. What if the ramp were also rough? How would we incorporate that friction?
3. How would you calculate the stopping distance if the friction coefficient were known?
4. How would the answer change if the box had an initial velocity along the horizontal surface?
5. What are some practical examples of calculating friction coefficients in everyday scenarios?

Tip: In energy conservation problems, always identify forms of energy present at each point and any non-conservative forces (like friction) that could affect total energy.