Force in the legs of a hexagonal table

In summary, the problem of determining the forces in each leg of a hexagonal table under the influence of a single applied force is insufficiently constrained, as there are only four equations and six potential unknowns. Assumptions about elasticity and rigidness must be made in order to find a solution.
  • #1
guv
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Homework Statement
Imagine a table shaped as a regular hexagon with length ##a## as shown in the attached figure. The table itself is massless. The table has 6 legs supporting it at each vertex on a leveled ground. Table itself is also horizontal.

Use the bottom left of the hexagon as the origin with side ##a## pointing to the right. Now suppose a force F is applied at a point ##(x,y)## on the table, what would be the force in each leg supporting the table?

The problem can be dimensionalized so the length ##a## is not really important, one can pretend ##a=1##
Relevant Equations
$$\vec F_{net} = 0$$
$$\vec \tau = 0$$
We know that the net force on the table must be zero
$$\sum F_i = F$$

We know that the components of the torque with respect to the origin is also 0.
$$\sum \tau_x = 0$$
$$\sum \tau_y = 0$$
$$\sum \tau_z = 0$$

But the problem becomes insufficiently constrained that there are only 4 equations while there are 6 potential unknowns other than the trivial symmetric cases. How to solve the 6 different tension force in each leg for arbitrary position ##(x,y)## where the force ##F## is applied?

Thanks
 

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  • #2
guv said:
Homework Statement:: Imagine a table shaped as a regular hexagon with length ##a## as shown in the attached figure. The table itself is massless. The table has 6 legs supporting it at each vertex on a leveled ground. Table itself is also horizontal.

Use the bottom left of the hexagon as the origin with side ##a## pointing to the right. Now suppose a force F is applied at a point ##(x,y)## on the table, what would be the force in each leg supporting the table?

The problem can be dimensionalized so the length ##a## is not really important, one can pretend ##a=1##
Relevant Equations:: $$\vec F_{net} = 0$$
$$\vec \tau = 0$$

We know that the net force on the table must be zero
$$\sum F_i = F$$

We know that the components of the torque with respect to the origin is also 0.
$$\sum \tau_x = 0$$
$$\sum \tau_y = 0$$
$$\sum \tau_z = 0$$

But the problem becomes insufficiently constrained that there are only 4 equations while there are 6 potential unknowns other than the trivial symmetric cases. How to solve the 6 different tension force in each leg for arbitrary position ##(x,y)## where the force ##F## is applied?

Thanks
The only way to get an answer to such a problem is to make assumptions about elasticity. Easiest would be to take the table as perfectly rigid and allow the floor to be compressed slightly, with equal spring constant and independently at each leg.
 
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Related to Force in the legs of a hexagonal table

1. What is force in the legs of a hexagonal table?

Force in the legs of a hexagonal table refers to the amount of weight or pressure that is exerted on each of the six legs of the table. This force is typically caused by the weight of the table itself, as well as any objects placed on top of it.

2. How is force distributed in the legs of a hexagonal table?

The force in the legs of a hexagonal table is evenly distributed among all six legs. This means that each leg is supporting an equal amount of weight, resulting in a stable and balanced table.

3. Can the force in the legs of a hexagonal table be affected by uneven surfaces?

Yes, the force in the legs of a hexagonal table can be affected by uneven surfaces. If the table is placed on an uneven surface, some legs may bear more weight than others, causing the table to wobble or become unstable.

4. How does the material of the table legs affect the force?

The material of the table legs can affect the force in different ways. For example, if the legs are made of a strong and sturdy material such as metal, they may be able to support more weight without bending or breaking. However, if the legs are made of a weaker material like wood, they may not be able to withstand as much force.

5. Is there a maximum amount of force that the legs of a hexagonal table can withstand?

Yes, there is a maximum amount of force that the legs of a hexagonal table can withstand. This is determined by the strength and durability of the materials used to make the legs, as well as the design and construction of the table itself. Exceeding this maximum force can cause the table to collapse or break.

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