How close to the edge of a table can a person sit without the table tipping over in the following scenario?
The table top weighs 20.0 kg in weight and the person weighs 65.0 kg. The table top is 2.30 m wide and 0.80 m from the ground. 1.30 m in length lies between the table legs and 0.50 m in length lies between the table's leg and the end of table on each side.
We assume that the person is seated at a distance d from the edge of the table. Now we can look at the table in the following way: The table top is a line with a pivot at the point where the leg meets the table top. The weight of the table is placed uniformly on the line and provides a torque about the pivot. The counteracting torque is provided by the weight of the person. For the table to not tip over, the torque due to the weight of the person should be equal to the torque due to the weight of the table top.
As the person is at a distance d from the edge, the torque exerted is equal to (0.5 - d)*65 + Int [ (20/2.3)x dx], x=0 to x=0.5.
We need to take the integral as the weight of the table top is uniformly distributed and at each point a torque is provided by it. Also, as g is present on both sides of the equation it gets canceled and is therefore ignored.
This is equal to 0.5*65 - 65d + (20/2.3)(x^2/2), x=0 to x= 0.5
=> 0.5*65 - 65d + (10/2.3)(.5^2 - 0).
On the other side of the pivot, the torque due to the weight of the table top is Int [ (20/2.3)x dx],x = 0 to x = 1.8
=> (10/2.3)[1.8^2 - 0]
Equating the two, we get
0.5*65 - 65d + (10/2.3)(.5^2 - 0) = (10/2.3)[1.8^2 - 0]
=> 0.5*65 - 65d + (10/2.3)(.25) = (10/2.3)(3.24)
=> 65d = (10/2.3)(2.5)+ .5*65 - (10/2.3)(3.24)
=> 65d = 29.28
=> d = 29.28/65
=> d = .45 m
So the closest distance from the edge that the person can sit is 45 cm. If the person moves closer to the edge, the table will tip over.