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The center of gravity is the point at which the total weight of a body or system is considered to be concentrated. In scenarios involving multiple forces, such as two people carrying a motor on a board, determining the center of gravity helps in understanding how the forces affect motion and balance.
For instance, in the exercise, finding the motor's center of gravity involves using moments, or the rotational equivalent of force. Simply put, if you take moments about one end of the board, the center of gravity is the point where these moments balance out.
In the given problem, for part (a), the board is lightweight, and thus, only the motor's weight needs consideration. The 600 N force creates a moment about the 400 N end, which helps locate the center of gravity 1.2 m from the 400 N end. In part (b), with the board weighing 200 N, its own center of gravity influences the overall position. This results in the center of gravity shifting slightly to 1.25 m from the 400 N end. This adjustment is because the board's weight now contributes to the total moment.

The moment of force, often referred to as torque, is the tendency of a force to rotate an object about an axis or pivot. It is calculated as the product of the force and the distance from the pivot or reference point to the line of action of the force.
When aiming to maintain static equilibrium, it is critical to balance these moments. In the exercise, you apply this principle by calculating moments to determine the center of gravity.
For part (a) of the problem, at the 400 N end, the moment caused by the 600 N force must be equal and opposite to the moment due to the motor's weight about the same point. By setting up the equation \( 600 \text{ N} \times 2 \text{ m} = 1000 \text{ N} \times x \), you solve for \( x \) to find the motor's center of gravity.
In part (b), with the board having its own weight, a new moment is introduced. Therefore, when calculating, you also add the board's contribution, leading to the equation: \( 600 \text{ N} \times 2 \text{ m} = 800x + 200 \text{ N} \times 1 \text{ m} \). Solving this gives a slightly different center of gravity value.

Calculating the total weight is essential in identifying equilibrium conditions. In the context of this problem, you need to calculate the weight of the motor when supported by forces at both ends of the board.
For part (a), because the board itself is light and negligible in weight, the total upward force exerted by both individuals on the motor is equal to the motor's weight. Therefore, summing the forces, \( 400 \text{ N} + 600 \text{ N} \), results in the motor's weight being 1000 N.
In part (b), however, the scenario involves a heavier board. The combined weight of the board and motor must be equal to the total 1000 N of the upward force provided by the people. Subtracting the board's weight of 200 N from the total suggests the motor itself weighs 800 N. This kind of calculation is crucial when dealing with static systems to ensure all forces balance accurately.