**Example 4.11 A Block on a Ramp** **Goal** Apply the concept of static friction to an object resting on an incline. **Problem** Suppose a block with mass 2.20 kg is resting on a ramp. If the coefficient of static friction between the block and the ramp is 0.450, what maximum angle can the ramp make with the horizontal before the block starts to slip down? **Strategy** This is an application of Newton's second law involving an object in equilibrium. Choose tilted coordinates, as in Figure 4.21. Use the fact that the block is just about to slip when the force of static friction takes its maximum value, \( f_s = \hat{\mu}_s n \). **Figure 4.21 Explanation** The diagram shows a block on a ramp inclined at an angle \( \theta \) to the horizontal. Vectors on the diagram include: - \( \vec{f}_s \) pointing up the ramp, representing static friction. - \( \vec{n} \) perpendicular to the surface of the ramp, representing the normal force. - \( mg \cos \theta \) perpendicular down. - \( mg \sin \theta \) parallel downward along the ramp, representing components of gravitational force. **Solution** 1. Write Newton's laws for a static system in component form. The gravity force has two components, just as in Examples 4.6 and 4.8. \[ \Sigma F_x = mg \sin \theta - \hat{\mu}_s n = 0 \quad (1) \] \[ \Sigma F_y = n - mg \cos \theta = 0 \quad (2) \] 2. Rearrange expression (2) to get an expression for the normal force \( n \). \[ n = mg \cos \theta \] 3. Substitute the expression for \( n \) into Equation (1) and solve for \( \tan \theta \). \[ \Sigma F_x = mg \sin \theta - \hat{\mu}_s mg \cos \theta = 0 \rightarrow \tan \theta = \hat{\mu}_s \] 4. Apply the inverse tangent function to get the answer. \[ \tan \theta = 0.450 \rightarrow \theta = \ **Remarks:** It's interesting that the final result depends only on the coefficient of static friction. Notice also how similar Equations (1) and (2) are to the equations developed in Examples 4.6 and 4.8. Recognizing such patterns is key to solving problems successfully. --- **Exercise 4.11** *Hints: Getting Started | I’m Stuck* The ramp in Example 4.11 is roughed up and the experiment repeated. (a) What is the new coefficient of static friction if the maximum angle turns out to be 40.5°? \[ \hat{\mu}_s = \boxed{0.65} \quad \textred{✕} \] Your response differs from the correct answer by more than 10%. Double check your calculations. (b) Find the maximum static friction force that acts on the block. \[ f_s = \boxed{\phantom{00}} \, \text{N} \]