A \(100 .-\mathrm{kg}\) uniform bar of length \(L=5.00 \mathrm{~m}\) is attached to a wall by a hinge at point \(A\) and supported in a horizontal position by a light cable attached to its other end. The cable is attached to the wall at point \(B\), at a distance \(D=2.00 \mathrm{~m}\) above point \(A\). Find: a) the tension, \(T,\) on the cable and b) the horizontal and vertical components of the force acting on the bar at point \(A\).

A free body diagram (FBD) is a powerful tool used in physics to illustrate the forces acting on an object. When drawing an FBD, you represent the object by a simple shape, usually a rectangle or dot, and then use arrows to depict the forces. Each arrow represents a force with its length proportional to the force's magnitude and direction that the force is applied.

For a bar suspended by a hinge and cable, like in our exercise, you would include forces such as the tension in the cable (\(T\)), the weight of the bar (\(W\)), which acts downward due to gravity, and the force at the hinge point (\(F_h\) for horizontal and \(F_v\) for vertical components). Understanding the FBD is crucial because it sets the stage for applying the conditions of equilibrium to solve for unknown forces.

The concept of equilibrium is central to understanding many physical systems. A body is in equilibrium when all the forces and torques acting on it balance out, resulting in no acceleration. There are two main conditions for equilibrium: the sum of all forces in each direction must be zero, and the sum of all torques about any point must also be zero.

In the context of our problem, torques are created by forces acting at a distance from the pivot point (\(A\)), creating a rotational effect. The equation \(WL/2 - TD\cos(\theta) = 0\) represents the torque balance where the weight's torque (\(WL/2\) due to its action at the bar's midpoint) is balanced by the torque produced by the tension in the cable (\(TD\cos(\theta)\)). When these torques are equal and opposite, the bar remains in a horizontal and stable position, fulfilling the condition for rotational equilibrium.

Tension is the force conducted along the length of a cable or string when it is pulled tight by forces acting from opposite ends. The tension in a cable can have vertical and horizontal components, which are of particular interest when the cable supports an object in equilibrium.

In the exercise, we use trigonometric functions to decompose the tension (\(T\) into its horizontal (\(T\cos(\theta)\)) and vertical (\(T\sin(\theta)\)) components. These components are crucial to satisfying the horizontal and vertical equilibrium conditions. Notably, the actual tension in the cable must be calculated taking into account the angle (\(\theta\) at which the cable is inclined to the horizontal, as seen in the step involving torque balance. Correctly identifying and calculating tension is key to solving many physics problems involving forces and can be especially tricky when multiple cables or angles are involved.