To solve for the coefficient of static friction ($\mu_s$) between the box and the ramp, we can use the angle at which the box just begins to slide down.

Step 1: Determine the Angle of the Incline

We have a right triangle with:

• Opposite side (height) = 60 cm
• Hypotenuse (ramp length) = 100 cm

The angle $\theta$ of the incline can be found using: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{60}{100} = 0.6$ Thus, $\theta = \arcsin(0.6) \approx 36.87^\circ$

Step 2: Relate the Angle to the Coefficient of Static Friction

At the point where the box just begins to slide, the static friction force is at its maximum and is equal to the component of gravity parallel to the ramp. The coefficient of static friction is given by: $\mu_s = \tan \theta$ Substituting $\theta \approx 36.87^\circ$: $\mu_s = \tan(36.87^\circ) \approx 0.75$

Conclusion

The coefficient of static friction between the box and the ramp is approximately 0.75.

Would you like further details or have any questions?

Here are five related questions you might find useful:

1. How is the frictional force calculated on an inclined plane?
2. What would happen if the angle of inclination were increased or decreased?
3. How does static friction differ from kinetic friction on an inclined plane?
4. How would you calculate the angle of an inclined plane if given only the coefficient of static friction?
5. What are the effects of different materials on the coefficient of friction?

Tip: When calculating angles with trigonometric functions, ensure your calculator is in the correct mode (degrees or radians) based on the problem’s requirements.