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# Determine the balanced state

This exercise is about the following questions:

• Which forces occur at frictionless contact points?
• How do you establish equilibrium conditions?
• How do you calculate lever arms for forces on inclined objects?

An ideally smooth rod with its own weight G should be inserted into a slot with the width B, so that the contact shown in A and B results and the state of equilibrium is established. What length l must the rod have so that it is in equilibrium at the angle Alpha?

## Solution

The following video is in German language, but English subtitles are available.

The way the problem is solved in the video is a little bit easier than the steps shown below.

The property “frictionless” results in the following reaction forces in the points A and B.

Balance of forces in x-direction:

$\tag{1} 0={F_A} \sin{\left( \alpha \right) }-{F_B}$

Balance of forces in y-direction:

$\tag{2} 0={F_A} \cos{\left( \alpha \right) }-G$

Balance of moments around A

$\tag{3} 0=G\, \left( \frac{l \cos{\left( \alpha \right) }}{2}-b\right) -\frac{{F_B} b \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) }}$

Solve for l

$\tag{4} l=\frac{2 {F_B} b \sin{\left( \alpha \right) }+2 G b \cos{\left( \alpha \right) }}{G\, {{\cos{\left( \alpha \right) }}^{2}}}$

Solving equations 1 and 2 for FA and FB

$\tag{5} {F_B}={F_A} \sin{\left( \alpha \right) }$

$\tag{6} {F_A}=\frac{G}{\cos{\left( \alpha \right) }}$

Substitution of FA

$\tag{7} {F_B}=\frac{G \sin{\left( \alpha \right) }}{\cos{\left( \alpha \right) }}$

Substitution of FB in equation 4

$\tag{8} l=\frac{\frac{2 G b\, {{\sin{\left( \alpha \right) }}^{2}}}{\cos{\left( \alpha \right) }}+2 G b \cos{\left( \alpha \right) }}{G\, {{\cos{\left( \alpha \right) }}^{2}}}$

Simplified:

$\tag{9} l=\frac{2 b\, {{\sin{\left( \alpha \right) }}^{2}}+2 b\, {{\cos{\left( \alpha \right) }}^{2}}}{{{\cos{\left( \alpha \right) }}^{3}}}$

Alternative notation of the result

$\tag{10} l=\frac{2 b\, \left( {{\tan{\left( \alpha \right) }}^{2}}+1\right) }{\cos{\left( \alpha \right) }}$