Find the minimum and maximum of a ratio for equilibrium

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SUMMARY

The discussion focuses on determining the minimum and maximum ratio of length to diameter (l/d) for a rod in equilibrium, considering static friction coefficients at points A and B. The friction force is defined as F = μ_s * F_n, where F_n represents the normal force. Participants are tasked with expressing the solution as an inequality involving μ_s and θ, specifically LS < l/d < RS. The calculations involve analyzing four forces and three torques in relation to the variables l, d, and θ.

PREREQUISITES
  • Understanding of static friction and its coefficient (μ_s).
  • Knowledge of normal force (F_n) and its role in equilibrium.
  • Familiarity with torque calculations and their application in mechanics.
  • Basic grasp of inequalities and their representation in mathematical expressions.
NEXT STEPS
  • Study the principles of static friction and how it affects equilibrium conditions.
  • Learn about torque calculations in rigid body mechanics.
  • Research methods for deriving inequalities in mechanical systems.
  • Explore the application of normal force in various equilibrium scenarios.
USEFUL FOR

This discussion is beneficial for students studying physics or engineering, particularly those focusing on mechanics and equilibrium analysis. It is also relevant for educators seeking to enhance their understanding of static friction and its implications in real-world applications.

Aleph_null
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Homework Statement



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Consider the rod as shown in the figure. The coefficient of static friction at A and B is \mu_s. I'm trying to find how big and how small the ratio l/d can be for the rod to still be in equilibrium. I have calculated how big all forces are acting upon the rod in the case just before sliding happens, but I'm not sure how to proceed.

Homework Equations



Just before sliding happens the friction force F is F = \mu_s * F_n, where F_n is the normal force.

The Attempt at a Solution

 
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They want the solution as an inequality in terms of \mu_s and theta, .i.e they want a LS and RS as LS < l/d < RS .
 
Aleph_null said:
I have calculated how big all forces are acting upon the rod in the case just before sliding happens, but I'm not sure how to proceed.

Have you? Could you show us that calculation? Did it involve 4 forces and 3 torques all written in terms of l, d, and θ?
 

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