Final Answer:
The force required at point A for the gate to remain closed is determined by the equilibrium of forces and is equal to the horizontal component of the tension in the cable, which is Tcos(θ).
Step-by-step explanation:
To understand the force required at point A, we consider the forces acting on the gate in equilibrium. In this scenario, the tension in the cable plays a crucial role. Let T represent the tension in the cable and θ be the angle between the cable and the horizontal axis. The force exerted at point A is the horizontal component of this tension, given by Tcos(θ).
In the equilibrium state, the sum of forces in the horizontal direction is zero. The force at point A is balanced by the horizontal component of the tension. Mathematically, this is expressed as ΣFx = 0, where ΣFx is the sum of forces in the x-direction. By applying trigonometric principles, we find that Tcos(θ) is the force exerted horizontally at point A.
The calculation involves understanding the geometry of the system and applying basic trigonometric functions. The force at point A is a result of the tension in the cable being effectively transmitted in the horizontal direction. This equilibrium condition ensures that the gate remains closed, and the force required at point A is precisely Tcos(θ).