# To find the angle between the ground and rods in limiting friction

• gnits
In summary: That clears everything up. In summary, the original method had ambiguities related to the mixing of limiting cases, while the new method does not.
gnits
Homework Statement
To find the angle between the ground and rods when in limiting friction
Relevant Equations
force balancing and moments

Here's my diagram:

The forces at the hinge (green) are internal forces.

For the whole system resolving vertically gives:

R1 + R2 = 4W

and horizontally gives:

F1 = F2

For the rod of weight 3W only, taking moments about B gives:

F1 * L * sin(Ѳ) + 3W * (1/2) * L * cos(Ѳ) = R1 * L * cos(Ѳ)

When in the limiting situation I can replace F1 with (2/3) * R1 and rearrange to give:

(2/3) * tan(Ѳ) = 1 - ( 3W / (2 * R1) )

For the rod of weight W only, taking moments about B gives:

R2 * L * cos(Ѳ) = F2 * L * sin(Ѳ) + W * (L/2) * cos(Ѳ)

When in the limiting situation I can replace F2 with (2/3) * R2 and rearrange to give:

(2/3) * tan(Ѳ) = 1 - ( W / (2 * R2) )

Now, and maybe this is bad reasoning?, we always have that F1 = F2, so in the limiting situation when either F1 = (2/3) * R1 or F2 = (2/3) * R2 then we will have that R1 = R2.

So, from the above equations, replacing R2 with R1, I will have:

For rod of weight 3W:

(2/3) * tan(Ѳ) = 1 - ( 3W / (2 * R1) )

and for rod of weight W:

(2/3) * tan(Ѳ) = 1 - ( W / (2 * R1) )

And the latter is the larger value and so the latter has the larger value of Ѳ in the limiting case and so rod BC will be the one which slips.

This agrees with the book answer.

However, for the next part, determining the angle, I do not get the book answer of 45 degrees.

Here I equate my two equations to give:

1 - (3*W) / (2 * R1) = 1 - W / (2 * R2)

which gives:

R1 = 3 * R2

and using R1 = 4W - R2 leads to R2 = W

Which, if put back into the expression for (2/3) tan(Ѳ) above, leads to tan(Ѳ) = 3/4

which does not imply that Ѳ = 45 degrees.

Where did I go wrong?

Thanks for any help.

gnits said:
then we will have that R1 = R2.
This is only the case if both limiting situations apply at the same time.
But it is very likely they do not.

##\ ##

Thanks for your reply. I have solved it now by starting over with another, less complicated route.

Lnewqban and BvU
gnits said:
Thanks for your reply. I have solved it now by starting over with another, less complicated route.
Could you show us that less complicated route?

Sure. My simpler method was to consider the system as a whole and take moments about A and then about C, these lead straight to R2 = 3W/2 and R1 = 5W/2. Then, looking at the conditions for no slipping at A and then C leads to F1 <= 5W/3 and F2 <= W and so slipping will occur at C as frictional forces will reach W before 5W/2. Finally, considering the rod of weight W only, take moments about B and this leads to 2 * F2 * tan(Ѳ) = 2 * R2 - W and in the limiting case we know F2 and R2 in terms of W and so can solve for Ѳ and this gives Ѳ = 45 degrees. My original method, even when corrected for the problem that was pointed out, still had ambiguities related to mixing of limiting cases at A and C, this new method does not suffer from this.

Lnewqban
gnits said:
Sure. My simpler method was to consider the system as a whole and take moments about A and then about C, these lead straight to R2 = 3W/2 and R1 = 5W/2. .
gnits said:
. My original method, even when corrected for the problem that was pointed out, still had ambiguities related to mixing of limiting cases at A and C, this new method does not suffer from this.
Thank you very much, gnits.

## 1. What is limiting friction?

Limiting friction is the maximum amount of friction that can occur between two surfaces before one starts to slide or move.

## 2. How do you calculate the angle between the ground and rods in limiting friction?

The angle between the ground and rods in limiting friction can be calculated using the formula tanθ = μ, where μ is the coefficient of friction between the two surfaces.

## 3. What factors affect the angle of limiting friction?

The angle of limiting friction is affected by the coefficient of friction, the weight of the object, and the surface area of contact between the two surfaces.

## 4. How does the angle of limiting friction impact the stability of an object?

The angle of limiting friction determines the maximum slope or angle at which an object can remain stationary without sliding. If the angle is exceeded, the object will start to slide due to the force of gravity.

## 5. Can the angle of limiting friction be changed?

Yes, the angle of limiting friction can be changed by altering the coefficient of friction between the two surfaces or by changing the weight or surface area of the object.

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