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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?

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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: $$Md_1=md_2$$

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  • $\begingroup$ Thank you! However, in the question I am doing the plank is 1.8m in total and the part of it that is over the edge is 1.4m so therefore there is only 0.4m of the plank laying on the cliff. I tried using your method and it is not working out as will the weight of the plank be over the edge? And therefore don’t know what d1 will be (the plank has a mass of 16kg?) $\endgroup$ – Molly Nov 30 '18 at 14:28
  • $\begingroup$ Ok, so the length of your plank is 1.8 m so it's center of mass is 0.9m from its any edge(1.8/2=0.9). Notice that the part of the rod that is on the cliff is 1.8-0.4=1.4m. So you see then d1=1.4-0.9=0.5m. Draw the diagram yourself it's easy then. $\endgroup$ – Tausif Hossain Nov 30 '18 at 15:19
  • $\begingroup$ Yes I got the 0.5 m and sub and re arranged the equation however I got the mass as 5.17kg which I think is the maximum mass but the question asks for the minimum mass? Which in the answers is 20kg? $\endgroup$ – Molly Nov 30 '18 at 16:18
  • $\begingroup$ If you use this setup you should get 20kg. Like: 160 x 0.5 = mg x 0.4, so m=20 kg. What is the answer given as? $\endgroup$ – Tausif Hossain Nov 30 '18 at 16:53
  • $\begingroup$ Thank you! So the answer is 20kg and that’s the exact set up as the demonstration equation in the answer. But why would you have 160 instead of 16? I presume that is the weight? But unsure why you would use 160? $\endgroup$ – Molly Dec 1 '18 at 4:29

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