Question

a wooden block of mass m is placed on a horizontal aluminum ramp. the angle of the ramp is slowly increased to 13.9° above the horizontal, at which point the block begins to slide down the ramp. determine the coefficient of static friction, μₛ, between the wooden block and the aluminum ramp. round your answer to two significant figures. reference information show calculator

Explanation:

Step1: Analyze Forces at Equilibrium

When the block is on the verge of sliding, static friction \( f_s \) is at its maximum (\( f_{s,\text{max}} = \mu_s N \)), and the forces along the ramp and perpendicular to the ramp are in equilibrium. Let the angle of the ramp be \( \theta = 13.9^\circ \). The gravitational force \( mg \) can be resolved into components: \( mg\sin\theta \) (parallel to the ramp) and \( mg\cos\theta \) (perpendicular to the ramp). Perpendicular to the ramp, \( N = mg\cos\theta \). Parallel to the ramp, \( f_{s,\text{max}} = mg\sin\theta \).

Step2: Substitute Maximum Static Friction

Since \( f_{s,\text{max}} = \mu_s N \) and \( N = mg\cos\theta \), substitute \( N \) into the friction equation: \( \mu_s mg\cos\theta = mg\sin\theta \).

Step3: Solve for \( \mu_s \)

Cancel \( mg \) from both sides: \( \mu_s \cos\theta = \sin\theta \). Then, \( \mu_s = \tan\theta \). Substitute \( \theta = 13.9^\circ \): \( \mu_s = \tan(13.9^\circ) \). Calculate \( \tan(13.9^\circ) \approx 0.248 \), round to two significant figures: \( \mu_s \approx 0.25 \).

Answer:

\( 0.25 \)