A 4.2 -m-long beam is supported by a cable at its center. A \(65-\mathrm{kg}\) steelworker stands at one end of the beam. Where should a \(190-\mathrm{kg}\) bucket of concrete be suspended for the beam to be in static equilibrium?

When dealing with the static stability of objects, such as beams, it's critical to understand the concept of moments about a pivot. In simple terms, a moment is the turning effect produced by a force applied at a distance from a pivot point. It is calculated as the product of the force and the distance from the pivot to the point where the force is applied, which is known as the moment arm. To visualize this, imagine pushing a door open; the force of your push multiplied by the distance from the hinges (the pivot) where you push is the moment causing the door to rotate.

In the context of the exercise, the beam acts as a see-saw with the cable at its center serving as the pivot. The steelworker and the bucket generate their own moments by applying force due to gravity at their respective distances from the pivot. To achieve equilibrium, the clockwise and counterclockwise moments about the pivot must be equal, so their combined rotational effects cancel out, preventing the beam from tipping over.

Torque balance, a concept closely related to moments, is a fundamental principle in statics that states that for a system to be in static equilibrium, all the torques acting on it must cancel out. In other words, the sum of clockwise torques must be equal to the sum of counterclockwise torques. Torque, often used interchangeably with moment, can be thought of as the rotational equivalent of force.

In our problem, the symmetrical suspension of the beam creates a scenario where the torque produced by the weight of the steelworker at one end must be balanced by the torque from the weight of the bucket of concrete. This principle allows us to set up an equation representing the system’s state of equilibrium, which we can solve to find the unknown position of the bucket. Understanding how to apply torque balance is essential to solving many real-world engineering problems involving statics.

Equilibrium calculations are used to determine the conditions under which an object or system will remain in a state of rest or move uniformly. For an object in static equilibrium, not only should the sum of all forces acting on it be zero, but also the sum of all torques. These calculations typically involve setting up equations based on the understood physical conditions, such as the torques being generated around a pivot.

For the exercise, the equation was set up using the weights of the steelworker and the bucket (as forces due to gravity) and their distances from the pivot point (acting as moment arms). By establishing that the moments from the steelworker and the bucket must be equal but opposite to maintain equilibrium, we can solve for the unknown distance for the bucket. This type of calculation is a cornerstone of engineering and physics, ensuring structures are stable and machinery functions correctly.
For quick reference, the steps in equilibrium calculations often involve:

• Identify all forces acting on the system.
• Determine the pivot point around which moments are calculated.
• Set up an equation equating the sum of clockwise moments to the sum of counterclockwise moments.
• Solve the resulting equation to find the unknowns needed for maintaining equilibrium.