The horizontal beam in Fig. E11.14 weighs 190 N, and its center of gravity is at its center. Find the horizontal and vertical components of the force exerted on the beam at the wall.

검증된 단계별 안내

Identify the forces acting on the beam: the weight of the beam (190 N) acting at its center, the tension in the cable, and the reaction forces at the wall (horizontal and vertical components).

Set up the equilibrium conditions for the beam. Since the beam is in static equilibrium, the sum of forces in both the horizontal and vertical directions must be zero, and the sum of torques about any point must also be zero.

Choose the point where the beam is attached to the wall as the pivot point to calculate torques. This will eliminate the reaction forces at the wall from the torque equation, simplifying the calculation.

Write the torque equation about the pivot point: \( \tau = 0 = (300 \text{ N} \times 4.00 \text{ m}) + (190 \text{ N} \times 2.00 \text{ m}) - T \times 3.00 \text{ m} \), where T is the tension in the cable.

Solve the torque equation for the tension T in the cable. Then, use the equilibrium conditions for forces to find the horizontal and vertical components of the reaction force at the wall. The horizontal component is equal to the horizontal component of the tension, and the vertical component is the sum of the vertical component of the tension and the weight of the beam.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rotational Equilibrium

Rotational equilibrium occurs when the sum of all torques acting on a system is zero, ensuring the system is not rotating. For a beam hinged to a wall, this means the torques due to the beam's weight and any other forces must balance. Calculating these torques involves considering the force magnitudes, their directions, and the distances from the pivot point.

Components of Force

Forces can be broken down into horizontal and vertical components, which are essential for analyzing equilibrium in two dimensions. The horizontal component acts parallel to the ground, while the vertical component acts perpendicular. In this problem, the force exerted by the wall on the beam can be resolved into these components to ensure both translational and rotational equilibrium.

Center of Gravity

The center of gravity of an object is the point where its weight is evenly distributed in all directions. For a uniform beam, this point is at its geometric center. Knowing the center of gravity is crucial for calculating the torque due to the beam's weight, as it determines the lever arm distance from the pivot point, affecting the rotational equilibrium.