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- Homework Statement
- A thin rod of mass m is welded on inner surface of a thin-walled pipe of mass M parallel to the axis of the pipe. The composite body thus formed is placed on a uniform slope as in the figure. Inclination of the slope is adjustable. Find range of coefficient of friction between the pipe and the slope for which the body slides down without rotation.

- Relevant Equations
- Torque

Here is the hint that the book gave me:

"For the maximum value of μ, the rod must be to the extreme right i.e. horizontally rightwards of the axis of the pipe"

I think what it meant is the same as this:

Note: in the calculation below, ##r## is the distance from the center of pipe to the CoM of the system and ##r = mR/(m+M)##

So, the torque calculation (with the system's CoM as origin) for maximum μ will be :

##N r = f R##

##Nr=NμR##

##m/(m+M)=μ##

But, the book says that it is ##m/\sqrt{M(M+2m)}##

Where did I do wrong?