- #1
PhysicsBoi1908
- 50
- 12
- Homework Statement
- A uniform right angled triangular lamina is placed on a horizontal floor, which is not frictionless. One of the acute angles of lamina is theta. If F_A and F_B are the minimum forces required to rotate the lamina about stationary vertical axes through A and B respectively, then find the minimum force required to rotate the lamina about a stationary vertical axis passing through C.
- Relevant Equations
- τ=Iα
![AD2D3081-3C0A-4F72-AC5D-6C9BF09237E1.jpeg AD2D3081-3C0A-4F72-AC5D-6C9BF09237E1.jpeg](https://www.physicsforums.com/data/attachments/249/249147-570c8822c2261a22ca24abd00ccb5c0f.jpg)
When the lamina rotates about A, FA must act on B (because it is the farthest away) perpendicular to AB (so that all of FA contributes to rotation).
Same argument is valid for rotation of lamina about B as well.
Having noted that, I tried two approaches:
Approach 1-
![9B357A72-7A5F-4340-AA3C-1B52BC64DB55.jpeg 9B357A72-7A5F-4340-AA3C-1B52BC64DB55.jpeg](https://www.physicsforums.com/data/attachments/249/249149-0f1551c52ebc856606707ea7186c6e8b.jpg)
If I assume that the lamina has mass m, then maximum static friction becomes μmg. FC must act on A such that it is perpendicular to AC. Then I just have to equate FClcosθ=∫dfr, where l is the length of the hypotenuse.
I can find df by writing dfr=dmαr
where FClcosθ=ICα, where IC is the moment of inertia of the lamina about C.
There are a lot of problems with this approach:
I don't know IC, I couldn't calculate it.
If I evaluated FC this way, then I would have to repeat the process for FA and FB as well, which would make the solution very lengthy.
Approach 2-
![AF67B420-5EB0-45C8-993E-C31526B40DC3.jpeg AF67B420-5EB0-45C8-993E-C31526B40DC3.jpeg](https://www.physicsforums.com/data/attachments/249/249148-ec1123325d887b4acc64c944e8b388c6.jpg)
While the last approach was, according to me, theoretically correct, I can't assure that for this one.
I argue that the minimum force required to turn A and B (and thus the lamina) about C must be the minimum force required to rotate lamina (and thus C) about A and B.
Then, the torque due to these forces about C must equal FClcosθ.
This approach dues give an answer, albeit the wrong one.
I think I know why the answer comes out to be incorrect. If FA or B is enough to cause rotation of A or B about C, then it can effectively rotate the lamina, and thus the answer must either by FA or B or something less than that.
Indeed, the correct answer is less than what I get from the above approach.
At this point, I have no more ideas.
(Note for approach one, I did find some videos online which derive moment of inertia of a triangular lamina, using "area moment" or something like that. But I would prefer to solve this question using high school physics, as the problem book expects me to.
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