Final answer:
To ensure a wood box slides down a vertical wall at constant speed, calculate the force applied at a 45° angle by considering the gravitational force and kinetic friction between the wood surfaces, and ensuring the net force is zero due to constant speed.
Step-by-step explanation:
To calculate the magnitude of the force you should apply to the wood box to make it slide down at a constant speed, you need to consider the forces acting on the box. These include the gravitational force, the force of friction, and the applied force. Since the box is sliding down at a constant speed, the net force acting on the box must be zero. This implies that the force of friction and the component of the gravitational force acting down the slope must be balanced by your applied force.
The gravitational force acting down the slope can be calculated using mg sin(θ), where m is the mass of the box, g is the acceleration due to gravity, and θ is the angle of the slope which is 45° in this case. The force of friction is given by μk mg cos(θ), with μk representing the coefficient of kinetic friction. To find the applied force, set the sum of forces parallel to the slope equal to zero (since the speed is constant) and solve for the applied force Fa. The applied force Fa has two components due to the 45° angle: one that counters the friction and the other that counters gravity.
Therefore, we can write the following equations: Fa cos(θ) - μk mg cos(θ) = 0 and Fa sin(θ) - mg sin(θ) = 0. By solving these equations, we find Fa = μk mg / (cos(θ) - sin(θ)). Given the mass of the box is 2.0 kg, the coefficient of kinetic friction is 0.200, and θ is 45°, we can calculate Fa. Remember to consider the correct units and use the approximation of g ≈ 9.8 m/s² for this calculation.