A shop sign weighing 215 N hangs from the end of a uniform 135-N beam as shown in Fig. 12–59. Find the tension in the supporting wire (at 35.0°), and the horizontal and vertical forces exerted by the hinge on the beam at the wall.

검증된 단계별 안내

Identify the forces acting on the system: The forces include the weight of the sign (215 N), the weight of the beam (135 N), the tension in the supporting wire (T), and the horizontal and vertical components of the hinge force (F_h and F_v).

Set up the torque equilibrium equation about the hinge: Since the system is in static equilibrium, the sum of torques about the hinge must be zero. Use the equation \( \sum \tau = 0 \), where \( \tau = r \cdot F \cdot \sin(\theta) \). Consider the torques due to the weight of the sign, the weight of the beam, and the tension in the wire.

Set up the force equilibrium equations: The system is also in translational equilibrium, so the sum of forces in both the horizontal and vertical directions must be zero. Use \( \sum F_x = 0 \) for horizontal forces and \( \sum F_y = 0 \) for vertical forces.

Solve for the tension in the wire (T): Use the torque equilibrium equation to isolate \( T \). Substitute the distances from the hinge to the center of mass of the beam and the sign, as well as the angle of the wire (35.0°).

Solve for the horizontal and vertical components of the hinge force: Use the force equilibrium equations to find \( F_h \) and \( F_v \) by substituting the value of \( T \) obtained from the previous step and balancing the forces in both directions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

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

Equilibrium of Forces

In physics, an object is in equilibrium when the net force acting on it is zero. This means that all the forces acting on the object must balance out. For the shop sign and beam system, this involves analyzing both the vertical and horizontal forces to ensure that they sum to zero, allowing us to determine the tension in the supporting wire and the forces at the hinge.

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (the hinge, in this case). In this scenario, we need to consider the torques created by the weight of the sign and the beam about the hinge to find the tension in the wire. The sum of the torques must also equal zero for the system to be in rotational equilibrium.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In this problem, the angle of the supporting wire (35.0°) will be used to resolve the tension into its vertical and horizontal components. Understanding how to apply these functions is crucial for calculating the forces acting on the beam and ensuring equilibrium.