Final answer:
To find the coefficient of static friction for a wooden block on a ramp, we compare the maximum static friction force to the gravitational force component parallel to the ramp. By using the relationship between these forces and the trigonometric functions for the given angle, the coefficient of static friction is calculated to be approximately 0.364.
Step-by-step explanation:
Finding the Coefficient of Static Friction
To determine the coefficient of static friction for a wooden block on a ramp, we first need to understand the forces at play. When the block is on the point of slipping, the force of static friction is at its maximum value and equal to the component of gravitational force parallel to the surface of the ramp. For the given problem, we will consider the block starting to slide down when the angle of the ramp is 20.0°.
The component of gravity acting down the ramp for a mass m on an incline is m Í g Í sin(θ), where g is the acceleration due to gravity and θ is the angle of inclination. The normal force acting on the block is m Í g Í cos(θ). The force of static friction is f_s = μ_s Í N, where N is the normal force and μ_s is the coefficient of static friction. At the verge of motion, the force of static friction equals the component of gravitational force parallel to the ramp: μ_s Í m Í g Í cos(θ) = m Í g Í sin(θ).
We can cancel out m and g since they appear on both sides of the equation, leading to μ_s Í cos(θ) = sin(θ). Substituting the given values:
μ_s Í cos(20.0°) = sin(20.0°)
Using trigonometric values:
μ_s Í 0.9397 = 0.3420
So, the coefficient of static friction μ_s is:
μ_s = 0.3420 / 0.9397 ≈ 0.364
Therefore, the coefficient of static friction for a wooden block with a mass of 1.5 kg on a wooden ramp at 20.0° is approximately 0.364.