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A uniform rod is made to lean between a ...

A uniform rod is made to lean between a rough vertical wall and the ground. The coefficient of friction between the rod and the ground is `mu_(1)` and between the rod and the wall is `mu_(2)`. Find the angle at which the rod can he leaned without slipping.

Text Solution

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`f_(1)=mu_(1)N_(1), f_(2)=mu_(2)N_(2)` ………..i
`HN_(2)=f_(1) =mu_(1)N_(1)`……..ii
`N_(1)+f_(2)=mg`
in limiting case `f_(1)=mu_(1)V` and `f_(2)=mu_(2)N_(2)` Substituting the value of `f_(1)` and `f_(2)` in eqn i and ii we get `N_(2)=(mu_(1)mg)/(1+mu_(1)mu_(2))`
Taking torque about `A` and -putting it to be zero
`mgl/2costheta=f_(2)lcostheta+N_(2)lsintheta`
`mgcostheta=2mu_(2)N_(2)costheta+2N_(2)sintheta`
Put the vaslue of `N_(2)` and simplify to get
`tantheta=(1-mu_(1)mu_(2))/(2mu_(1))`
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