Condition for stable equlibrium

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Discussion Overview

The discussion revolves around determining the minimum radius of a semicircular cylinder required for a wooden bar to achieve stable equilibrium when balanced on top. It involves concepts of potential energy and equilibrium conditions, with participants exploring the mathematical relationships and physical principles involved.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant states that stable equilibrium requires the first derivative of potential energy to be zero and the second derivative to be positive, questioning their approach to expressing potential energy in terms of the radius R.
  • Another participant challenges the assumption that the center of mass (CM) remains at a fixed height of R+2 cm when the wooden bar is displaced, suggesting that the CM changes with displacement.
  • A participant calculates the new height of the CM after displacing the block by an angle θ, presenting a formula for height that includes terms dependent on R and θ, but questions the relevance of the bar's width in this context.
  • There is a light-hearted clarification regarding the terminology used, with a participant correcting the reference from "width" to "length" of the bar.

Areas of Agreement / Disagreement

Participants express differing views on the behavior of the center of mass during displacement and the relevance of the bar's dimensions, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

Participants have not fully resolved the implications of their calculations regarding the potential energy and equilibrium conditions, and there are indications of missing steps in the mathematical derivation.

RajarshiB91
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Homework Statement



A homogeneous wooden bar of length 10 cm, thickness 4 cm and weight 1 Kg is balanced
on the top of a semicircular cylinder of radius R as shown below. Calculate the
minimum radius of the semicircular cylinder if the wooden bar is at stable equilibrium.
?temp_hash=ff5d8793a573727d0f3d7c73d119c317.jpg


Homework Equations



Potential energy E=mgh and its derivatives.

The Attempt at a Solution



Stable equilibrium means the first derivative of potential energy is zero and its second derivative must be greater than zero(local minima). So, I have to express the PE of the wooden bar in terms of R and find minimum R to satisfy above conditions. But here the CM of the bar is at R+(4/2)=R+2 cm above the ground. So, second derivative of PE is always 0? Where am I going wrong? Also, how to approach problems like these in general? I had read about equilibrium a long time back and the concepts are a bit muddled up.
 

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The CM doesn't stay at R+2 if the beam is wiggled !
 
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BvU said:
The CM doesn't stay at R+2 if the beam is wiggled !
Thanks BvU. Yes, I googled similar problems and understood what needs to be done. By displacing the block by θ I calculate the new CM height which comes out to be
h=Rcosθ+Rθsinθ+2cosθ
Now, it is just differentiating twice. So, for equilibrium, is the width of the block(10 cm) irrelevant? I didn't find it's use in the height equation or am I missing something?
 
Ah, you mean the length :smile: !

Didn't work out h myself, and you don't show the steps, so I can't really tell. Suppose you're right.
 

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