A large 62.0 kg board is propped at a 44 angle against the edge of a barn door that is 3.1 wide.How great a horizontal force must a person behind the door exert (at the edge) in order to open it? Assume that there is negligible friction between the door and the board but that the board is firmly set against the ground.
To determine the horizontal force needed to open the door, we need to analyze the forces acting on the board. Here's how:
1. Identify the forces: In this scenario, there are two main forces acting on the board:
- The weight (mg) of the board acting downward.
- The normal force (N) exerted by the ground, perpendicular to the surface of the board.
2. Resolve the weight force: The weight of the board acts vertically downward. We need to resolve this force into its components parallel and perpendicular to the door. The parallel component is given by mg sinθ, where θ is the angle of the board.
3. Calculate the normal force: The normal force is equal in magnitude and opposite in direction to the parallel component of the weight force.
4. Calculate the force required to open the door: The force required to open the door is the horizontal component of the normal force. This force can be calculated using the equation F = N cosθ, where θ is the angle of the board.
Let's do the calculations step by step:
1. Weight force: The weight of the board is given by mg, where m = 62.0 kg and g = 9.8 m/s^2 (acceleration due to gravity).
Weight = 62.0 kg * 9.8 m/s^2 = 607.6 N
2. Parallel component of weight:
Parallel component = Weight * sinθ
Parallel component = 607.6 N * sin(44°) ≈ 400.65 N
3. Normal force:
Normal force = Parallel component of weight
Normal force = 400.65 N
4. Force required to open the door:
Force = Normal force * cosθ
Force = 400.65 N * cos(44°) ≈ 284.90 N
Therefore, a person behind the door must exert a horizontal force of approximately 284.90 Newtons (N) to open it.