Final answer:
To find the sliding friction force on the crate, use the worker's weight (which equals the tension in the rope) and subtract the gravitational force component pulling the crate down the plane. Once the frictional force is obtained, calculate the coefficient of kinetic friction by dividing this force by the normal force on the crate.
Step-by-step explanation:
The magnitude of the sliding friction force on the crate can be found by analyzing the forces acting on the crate and the worker. Since the worker with a mass of 57 kg is hanging his entire weight to move the crate up at a constant velocity, the force applied by the worker (which is the same as the tension in the rope) will equal the weight of the worker, F_t = m_worker * g, which equals 57 kg * 9.8 m/s2. Given the plane is inclined at 22°, we have to find the component of the gravitational force pulling the crate down the plane. This force is m_crate * g * sin(θ), where m_crate is the mass of the crate, g is gravitational acceleration, and θ is the angle of the inclination. The force of friction, F_friction, plus the component of gravitational force down the plane, must equal the tension in the rope since the crate is moving at constant velocity (meaning the net force is zero). Thus, F_friction = F_t - m_crate * g * sin(θ). Plugging in the values gives us the magnitude of the frictional force. Once we have the frictional force, we can find the coefficient of kinetic friction, μ_k, by the formula F_friction = μ_k * N, where N is the normal force acting on the crate. The normal force is equal to m_crate * g * cos(θ). The μ_k can thus be found by dividing the frictional force by the normal force and rounding to two decimal places.