A large 61.0 kg board is propped at a 45 degree angle against the edge of a barn door 2.9m 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 how great a horizontal force is required to open the door, we need to analyze the forces acting on the board.
The weight of the board is acting downward and can be calculated using the formula:
Weight = mass × gravitational acceleration.
Given that the mass of the board is 61.0 kg and the gravitational acceleration is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 61.0 kg × 9.8 m/s^2 = 598.8 N.
Now, let's break down the weight into its components. Since the board is propped at a 45-degree angle, we can resolve the weight into two parts: one acting perpendicular to the door and the other acting parallel to the door.
The perpendicular component of the weight (W_perpendicular) is given by the formula:
W_perpendicular = Weight × cos(angle).
The parallel component of the weight (W_parallel) is given by the formula:
W_parallel = Weight × sin(angle).
In this case, the angle is 45 degrees, so we can calculate the perpendicular and parallel components:
W_perpendicular = 598.8 N × cos(45°) ≈ 424.0 N.
W_parallel = 598.8 N × sin(45°) ≈ 424.0 N.
Since there is negligible friction between the door and the board, the only force that needs to be overcome is the perpendicular component of the weight, which tends to push the door closed.
Therefore, the person exerting the force at the edge of the door must apply a force equal in magnitude but opposite in direction to the perpendicular weight component. Therefore, the horizontal force required to open the door is approximately 424.0 N.