# Balanced Torques and centre of mass

Say we have a plank of wood that is placed on the edge of a cliff so that one end projects horizontally out over the edge. How would you go about finding the minimum mass that you could place on top of the plank so it doesn’t topple over the edge and remains in equilibrium.

I know the torques must be balanced, but how would the equation for the mass of the object be derived.

Also i think the object should be placed at the end of the plank furthest out from the overhang however what about the center of mass?

Consider a plank of mass $$M$$ kg and let the mass you need to hang be $$m$$ kg. For this system, the limiting case is when the normal reaction force acts at the edge of the cliff(as here the torque due to the weight of the plank just balances the torque due to the weight of the mass $$m$$). Hence just find the distance of the edge of the cliff from the center of the plank($$d_1$$) and the distance from the center of the mass $$m$$ to the edge of the cliff($$d_2$$). Then use the principle of moments: $$Md_1=md_2$$