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The center of gravity is the point where the total weight of an object or system can be considered to act. It is crucial for understanding how objects balance. In the context of this exercise, the center of gravity for the motor is initially determined by considering just the forces applied by the two people. This is because to find where the motor "balances" on the board, we need to know where its weight effectively pulls down.

For instance, when the motor is supported by these forces alone, its center of gravity is not at the midpoint of the board but rather at 1.20 m from the end where the lesser force is applied (400 N). This uneven distribution arises because different force magnitudes affect how the weight is spread along the board.

When the board's own weight is considered, the entire system's center of gravity changes. Now the board's weight distribution also factors into the balance, shifting the motor's center of gravity to a new point. This shows how additional weights and their placements can alter balance dynamics.

Torque is the measure of how much a force acting on an object causes it to rotate. Mathematically, it's calculated as the product of the force and the distance from the pivot point (torque arm). The equation for torque, \( \tau = F \cdot d \), highlights this relationship.

In this problem, to find the motor's center of gravity, we calculate the torques applied by the lifting forces. Taking torques about the board's end with the 400 N force, we set the following equilibrium condition \( 0 = 400 \times 0 + 600 \times 2.00 - W \times x \). Solving for \( x \) (the motor's center of gravity) shows how far from the pivot the motor 's center acts.

With the board added in the second part, the calculation shifts. The board's own weight introduces new torque, hence balancing torques now includes contributions from both the motor and board: \( 800x = 200 + 1200 \). Each element's position and weight adjusts how the system achieves equilibrium.

Force balance ensures that an object in equilibrium doesn't accelerate. This means the total forces pushing and pulling on it must cancel each other out. In this exercise, the forces from both people (400 N and 600 N) should sum up to match the downward force of the motor's weight. When they add up to 1000 N, it perfectly balances the motor against gravity.

When the board's weight of 200 N is factored in, it changes the force distribution without changing the total force applied by the lifters. This effectively means the motor's weight must adjust to 800 N so that the total weight, considering the board, still aligns with the overall lifting efforts. This principle is fundamental in physics for understanding balance and rotation.

The concept of weight is straightforward—it's simply the force of gravity acting on an object. However, understanding how this plays into equilibrium problems can be less intuitive.

In the problem given, the motor's weight is initially what balances the forces exerted by the two people. This direct calculation equates the sum of these forces (400 N + 600 N = 1000 N) to the motor's weight.

Adding the weight of the board (200 N) changes the scenario. Now, the combined weight of the motor and board still needs to be 1000 N to maintain balance given unchanged exertion by the carriers. Therefore, the motor's adjusted weight becomes 800 N to preserve this balance. This shift illustrates how distributions affect overall systems, an important concept in physics.