91Ó°ÊÓ

Understanding the center of gravity is crucial when analyzing static equilibrium problems. The center of gravity of an object is the point where the entire weight of the object can be considered to act. For uniform objects, like bridges or beams, the center of gravity is located at the geometric center.

In the original exercise, the bridge is assumed to be uniform, which means its center of gravity is at its midpoint, or halfway across the bridge. It's important to distinguish that even if a structure is symmetrically shaped, the actual center of gravity could shift if the weight is not evenly distributed across the structure.

• The weight of the bridge acts vertically downward from its center of gravity.
• The hiker's weight acts downward at a point one-fifth of the way from the start to the far end of the bridge.

Understanding the location of these weighted actions helps in developing the calculations for equilibrium and moments.

Moment of force, often referred to as torque, is a measure of the rotational effect of a force applied at a distance from a pivot point. This concept is key to solving equilibrium problems, as it helps determine how different forces create twists and rotations.

In the exercise, we take moments about the near end of the bridge to solve for the force exerted by the support at the far end.

• The counterclockwise moments are considered positive.
• The key forces producing moments are the bridge's weight at its center and the hiker's weight at one-fifth of the way along the bridge.

These distances from the pivot (near end) to the points where the forces act are crucial in calculating the moments. The balance of moments ensures that the bridge does not rotate, keeping it in equilibrium.

Vertical forces are the forces acting in the vertical direction along the vertical axis of an object in equilibrium. In problems dealing with equilibrium, understanding how these forces interact is vital.

For the bridge problem, the vertical forces include:

• The downward forces due to the weight of the hiker (985 N) and the weight of the bridge (3610 N)
• The upward forces are the reactions at the supports (R1 and R2)

The equilibrium principle asserts that the sum of the upward forces must equal the sum of the downward forces, maintaining equilibrium in the vertical direction:
\[ R1 + R2 = 4595 \] Where 4595 N is the total downward force acting on the system. This balance ensures that the bridge remains stable without any net upward or downward movement.

Equilibrium conditions are fundamental principles applying to any system in balance, ensuring that there is neither linear acceleration nor rotational movement. This means that both the sum of forces and the sum of moments (torques) must be zero.

For the bridge with the hiker, the equilibrium conditions lead us to two key equations:

• For vertical forces: \( R1 + R2 = 4595 \) N, ensuring no vertical motion.
• For moments about an axis: \( R2 \times L - 3610 \times \frac{L}{2} - 985 \times \frac{L}{5} = 0 \) ensures no rotational movement.

The equilibrium conditions are applied to determine the forces at each support, R1 and R2, by analyzing the sum of forces and moments. These principles ensure that the system remains stable, with no stairs or tipping off the supports.