Question

The top view of a table, with weight Wt, is shown in the figure. (Figure 1) The table has lost the leg at (Lx, Ly), in the upper right corner of the diagram, and is in danger of tipping over. Company is about to arrive, so the host tries to stabilize the table by placing a heavy vase (represented by the green circle) of weight Wv at ( X, Y). Denote the magnitudes of the upward forces on the table due to the legs at (0, 0), (Lx, 0), and (0, Ly) as F0, Fx, and Fy, respectively. Find Fx, the magnitude of the upward force on the table due to the leg at (Lx, 0). Express the force in terms of Wv, Wt, X, Y, Lx, and/or Ly. Note that not all of these quantities may appear in the answer.

Answer 1

The magnitude of the upward force, Fx, exerted by the leg at position (Lx, 0) on the table can be expressed as proportional to the weight of the vase, Wv, multiplied by the ratio of the horizontal distance between the missing leg and the vase (X) to the horizontal distance between the missing leg and the table's origin (Lx).

Fx = Wv * (X / Lx)

Step-by-step explanation:

To find the magnitude of the upward force Fx on the table due to the leg at (Lx, 0), we can analyze the equilibrium of forces acting on the table. Considering the moments about the point where the leg is missing, the table can be stabilized by the vase. This equilibrium can be represented as the sum of the moments caused by the forces. Given that the table is on the verge of tipping, the moment caused by the vase (Wv) at (X, Y) counters the missing leg's moment.

By using the principle of moments, the moment exerted by the vase (Wv) at position (X, Y) is equal to the moment caused by the missing leg at (Lx, 0). Hence, setting up the equation for equilibrium of moments, we derive that Fx, the force at (Lx, 0), is equal to Wv multiplied by the ratio of the distance X (from the missing leg to the vase) to the distance Lx (from the missing leg to the origin).

This equilibrium is crucial in maintaining stability and preventing the table from tipping over. By calculating the force at (Lx, 0) using this ratio of distances, it allows for a stable configuration ensuring the table remains balanced despite the missing leg, ensuring the safety and convenience of the arriving guests.