A large 62.5kg board is propped at a 42 degree angle against the edge of a barn door that is 2.7m 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 solve this problem, we need to consider the forces acting on the board. The main forces at play are the weight of the board and the force exerted by the person pushing the board.

First, let's analyze the weight of the board. The weight of the board can be calculated using the formula:

Weight = mass x gravity

Given that the mass of the board is 62.5 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight of the board:

Weight = 62.5 kg x 9.8 m/s² = 612.5 N

Now, let's analyze the forces acting along the board. We can decompose the weight of the board into its horizontal and vertical components. The vertical component will act perpendicular to the inclined surface of the board, and the horizontal component will act parallel to the surface of the board.

The vertical component of the weight can be calculated using the equation:

Vertical Component = Weight x sin(angle)

Given that the angle of inclination is 42 degrees, we can calculate the vertical component:

Vertical Component = 612.5 N x sin(42°) = 417.35 N

The horizontal component of the weight can be calculated using the equation:

Horizontal Component = Weight x cos(angle)

Calculating the horizontal component:

Horizontal Component = 612.5 N x cos(42°) = 464.06 N

Now, we need to consider the forces exerted by the person behind the door. Since there is negligible friction between the door and the board, the horizontal component of the force exerted by the person must balance out the horizontal component of the weight of the board.

Therefore, the horizontal force exerted by the person must be equal to the horizontal component of the weight of the board:

Horizontal Force = 464.06 N

Hence, the person behind the door must exert a horizontal force of 464.06 N to open the door.