Question

The frame shown in the figure is loaded at S with a force of 2000 N. The reactions at T are denoted by \( T_x \) and \( T_y \), while the reaction at W is \( W_y \). Neglect the weight of the members. Which one of the following options for the magnitudes of the forces (in N), \( T_x \), \( T_y \), and \( W_y \), is CORRECT?

• A. \( T_x = 0, T_y = 1000 \) and \( W_y = 1000 \)
• B. \( T_x = 0, T_y = 1500 \) and \( W_y = 500 \)
• C. \( T_x = 0, T_y = 800 \) and \( W_y = 1200 \)
• D. \( T_x = 0, T_y = 500 \) and \( W_y = 1500 \)

Hint:

When solving force equilibrium problems, remember that the sum of forces in both the horizontal and vertical directions must be zero. Break down the forces into their components and use the symmetry of the system for easier calculations.

Answer 1

The Correct Option is D

We are given that a 2000 N force is applied at point \( S \), and the frame is in equilibrium. The reactions at point \( T \) are denoted as \( T_x \) and \( T_y \), while the reaction at \( W \) is \( W_y \).
Step 1: Equilibrium Conditions
In static equilibrium, the sum of forces in both horizontal and vertical directions must be zero.
Horizontal Direction:
The force at point \( T \) only has a horizontal component, \( T_x \). Since there is no external horizontal force acting on the system, we have: \[ \sum F_x = 0 \Rightarrow T_x = 0. \]
Vertical Direction:
The vertical forces are the external force of 2000 N applied at \( S \), the vertical component of the reaction at \( T \) which is \( T_y \), and the vertical reaction at \( W \), \( W_y \). The sum of forces in the vertical direction must be zero: \[ \sum F_y = 0 \Rightarrow T_y + W_y = 2000. \] So, the sum of the vertical components of the reactions must equal the applied force at \( S \), which is 2000 N.
Step 2: Solving for \( T_y \) and \( W_y \)
To solve for \( T_y \) and \( W_y \), we analyze the geometry of the system. By using the equilibrium conditions and considering the symmetry of the frame, we can determine that the forces must be balanced in such a way that: \[ T_y = 500 \, {N}, W_y = 1500 \, {N}. \] This satisfies the equilibrium equation for vertical forces: \[ T_y + W_y = 2000 \Rightarrow 500 + 1500 = 2000. \] Thus, the correct answer is \( T_x = 0, T_y = 500 \) and \( W_y = 1500 \).