Physics Hardie?

Question

A 21 kg round table is supported by three legs equal distances apart on the edge. What minimum mass, placed on the table's edge, will cause the table to overturn?

Answer 1

It depends exactly where the weight is placed relative to the legs. If it's exactly halfway between the legs on the edge, then the following is true:
Picture an equal isoscoles triange (all angles are 60 degrees) inscribed in a circle. The legs are at the vertices of the triangle, while the circle is the tabletop. If the table had a clock face, the legs would be at 12, 4, and 8. Place the weight at 2 on the clock. Now you have a "fulcrum" represented by the line between the legs at 12 and 4. Now, draw a line from the center of the table (which is the center of mass of the table) to the weight. It will be perpendicular to the fulcrum line you've just drawn. A little geometry will tell you that the distance from the center of the table to the fulcrum line is the same as the distance from the fulcrum line to the weight. The table will overturn when the weight of the table times the distance from the center to the fulcrum is exceeded by the weight of the mass times the distance to the fulcrum. Since the distances are the same, the table will overturn when the mass exceeds the mass of the table - 21 kg.

If the mass isn't exactly halfway between the legs, the same technique applies - the geometry is just a little harder.

Answer 2

This problem is doable, but difficult.

First thing you have to do is pick which of the three sides you want to tip. The axis of rotation will run between two of the legs (any two will do).

Next you have to sum up all of the torques around that axis. That means multiplying the distance from the axis to the center of the table times the weight of the table (mass times g).

Once you do this, you know how much torque the weight will have to exert to tip the table.

Calculate the distance from the axis to the edge of the circle (requires some geometry skills).

Divide the torque by that distance to get the weight required.

Divide that weight by g to get the mass required.

Lots of steps, but it's doable.

Grizzly's answer is good--much cleaner if you have geometric insight. :)

Answer 3

One gram, if you let me design the table. Pick the place where you'll want to add that mass. Build the table with most of its mass along that edge, and just enough mass in the rest of the table to keep it overturning on its own.

Answer 4

Hi - I don't think it can be possible to answer this question without some more information - you would need to know the diameter of the table.

With this information, you should be able to work out the moment required to pivot the tabletop across two legs and overcome the weight of the table acting to hold it in place originally.

I may be wrong - I'd be interested to know how it's done if there is a way.