What is the maximum coefficient of friction for a composite body on a slope?

In summary: That's OK. One can find ##\mu## at some angle ##\varphi## below the horizontal line and then maximize ##\mu## with respect to ##\varphi##. Just a little bit more work, but that's what I did to explore what ##\mu## looks like as a function of ##\varphi.##
  • #1
Rikudo
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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
1657948211713.png


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:
1657948709558.png

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?
 
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  • #2
I would start with this picture
torque wrt CoM: torque from normal force = Nrsinθ = torque from friction force = μN*L
Find L as an expression of r and θ
1657953100131.png
 
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  • #3
drmalawi said:
The weight of the rod will also produce a torque
This will be neutralized by torque produced by the pipe's weight. Hence, in the torque equation, we only use friction and normal force.
 
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  • #4
Rikudo said:
This will be neutralized by torque produced by the pipe's weight. Hence, in the torque equation, we only use friction and normal force.
Yeah I forgot that it should only slip, not rotate at all.
Look at the pic above, try to calculate µ as a function of θ
 
  • #5
@Rikudo now picture is correct...
 
  • #6
AH! Now I got it. Thanks!
 
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  • #7
Rikudo said:
"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:
View attachment 304258
For information, the hint says:
"For the maximum value of μ, the rod must be to the extreme right i.e. horizontally rightwards of the axis of the pipe"​

But in your diagram, the rod (red dot) is not at the extreme right, and therefore not at the same horizontal level (i.e. not at the same height) as the axis of the pipe.
 
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  • #8
Steve4Physics said:
But in your diagram, the rod (red dot) is not at the extreme right, and therefore not at the same horizontal level (i.e. not at the same height) as the axis of the pipe.
That's OK. One can find ##\mu## at some angle ##\varphi## below the horizontal line and then maximize ##\mu## with respect to ##\varphi##. Just a little bit more work, but that's what I did to explore what ##\mu## looks like as a function of ##\varphi.##
 

1. What is a composite body on a slope?

A composite body on a slope refers to an object that is made up of multiple materials or components and is placed on an inclined surface.

2. How does the composition of a body affect its behavior on a slope?

The composition of a body can greatly influence its behavior on a slope. Different materials have varying densities, strengths, and frictional properties, which can affect how the body moves or slides down the slope.

3. What factors determine the stability of a composite body on a slope?

The stability of a composite body on a slope is determined by several factors, including the angle of the slope, the weight and distribution of the body, and the friction between the body and the slope.

4. What is the significance of studying a composite body on a slope?

Studying composite bodies on slopes is important in various fields, such as engineering and geology, as it helps us understand the behavior of different materials on inclined surfaces. This knowledge can be applied in designing structures or predicting natural phenomena, such as landslides.

5. How can we calculate the motion of a composite body on a slope?

The motion of a composite body on a slope can be calculated using principles of physics, such as Newton's laws of motion and the concept of friction. By considering the forces acting on the body, we can determine its acceleration and predict its motion on the slope.

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