Friction Problem in Statics

[lc99]

Homework Statement

Homework Equations

The Attempt at a Solution

I've spent so long trying to solve this system of equations with my calculator that can do this for me. But, somehow , i still don't get it.The equations i got are:
1st diagram
Fc - Nb =0
Nc -0.3Nb -300=0
1.5*Nb-300x-0.5(0.3)Nb - 0.3Nb*x =0

second diagram
Nb-0.2Na=0
Na - 0.3Nb -180=0
M-Nb(1.5)-0.3Nb(1.5) =0

Assuming that Fa = 0.2Na and Fb = 0.3Nb because slipping will occur at A and B.

I've entered these into system of equations solver, and I keeping gett Fa = 1800/47 = 38.3
Fb =11.49
Nb = 38.3

I plugged Nb=38.3 for the last equation, and got M = 74.7 but the answer is wrong! I am beyond frustrated. :(

 

[kuruman]

Is the concrete block going to tip before it slips or slip before it tips? How does the threshold for tipping compare against the threshold for slipping? What about the threshold for the wheel slipping?

 

[Nathanael]

You should think about kurumans good questions.

I just want to point out a small mistake;

second diagram
Nb-0.2Na=0
Na - 0.3Nb -180=0
M-Nb(1.5)-0.3Nb(1.5) =0

F_B = 0.3N_b is drawn upwards, so by your conventions it should be positive not negative.

Also all your equations should really be inequalities... it’s not necessarily true that all static frictional forces will be at their max.

 

[lc99]

You should think about kurumans good questions.

I just want to point out a small mistake;
F_B = 0.3N_b is drawn upwards, so by your conventions it should be positive not negative.

Also all your equations should really be inequalities... it’s not necessarily true that all static frictional forces will be at their max.

I'm sorry. I'm fairly new to this 'dry friction' topic and tipping vs slipping threshold. I do know that tipping occurs if the moment from pushing is greater tha the moment of friction. Also that slipping occurs when Force pushing is greater then friction. However, I'm sort of confused with how to analyze this with multiple friction involved.

So, I've literally just made assumptions. These questions I can answer, but It's hard to know where to start :(

 

[haruspex]

I'm sorry. I'm fairly new to this 'dry friction' topic and tipping vs slipping threshold. I do know that tipping occurs if the moment from pushing is greater tha the moment of friction. Also that slipping occurs when Force pushing is greater then friction. However, I'm sort of confused with how to analyze this with multiple friction involved.

So, I've literally just made assumptions. These questions I can answer, but It's hard to know where to start :(

First, you need to think of all the ways the system might be about to move. List them.
In each scenario, some frictional forces will still be below their max, while others may necesarily be simultaneously hitting their max. For each scenario, list the forces that will be at their limits. For each such force you can write an equation. Thus, for each scenario you will have a set of equations you can hope to solve.

It is not especially hard, just a matter of being organised, imaginative and patient.

 

[kuruman]

Adding to what @haruspex wrote, be sure to draw the tails of the arrows representing forces at the points where these forces act. It will make it easier to find the appropriate lever arms for your torque calculations.

 

[lc99]

First, you need to think of all the ways the system might be about to move. List them.
In each scenario, some frictional forces will still be below their max, while others may necesarily be simultaneously hitting their max. For each scenario, list the forces that will be at their limits. For each such force you can write an equation. Thus, for each scenario you will have a set of equations you can hope to solve.

It is not especially hard, just a matter of being organised, imaginative and patient.

Since there are 3 friction forces to be analyzed, I'm thinking that the scenerio that is most likely to take place is where the concrete block doesn't slip or tip.

First, you need to think of all the ways the system might be about to move. List them.
In each scenario, some frictional forces will still be below their max, while others may necesarily be simultaneously hitting their max. For each scenario, list the forces that will be at their limits. For each such force you can write an equation. Thus, for each scenario you will have a set of equations you can hope to solve.

It is not especially hard, just a matter of being organised, imaginative and patient.

First, you need to think of all the ways the system might be about to move. List them.
In each scenario, some frictional forces will still be below their max, while others may necesarily be simultaneously hitting their max. For each scenario, list the forces that will be at their limits. For each such force you can write an equation. Thus, for each scenario you will have a set of equations you can hope to solve.

It is not especially hard, just a matter of being organised, imaginative and patient.

Is this scenario possible where A slips but B doesn't and vise versa? I don't think it is, so I did not set up a scenario for that.

Also, in the scenario that A slips, B slips, and Concrete doesn't slip, is the location of normal force at C and A remain at the center of each object?

 

[lc99]

Is the concrete block going to tip before it slips or slip before it tips? How does the threshold for tipping compare against the threshold for slipping? What about the threshold for the wheel slipping?

Okay, so i think i figured out that the concrete block tips before slipping. The wheel slips at A and B before the concrete tips. How can i use this to find M now? In each of these cases, I didn't have an 'x' variable. Is this okay? Cause after i figured out that A and B slips before concrete tips, I used the condition that A and B slips to decrease the unknowns. Doing this, helped me solve for M.

So, in the end, the process of figuring out whether concrete block tips before slipping allows me to identify the situation that the problem is in?

 

[lc99]

Okay, so i think i figured out that the concrete block tips before slipping. The wheel slips at A and B before the concrete tips. How can i use this to find M now?

Is the concrete block going to tip before it slips or slip before it tips? How does the threshold for tipping compare against the threshold for slipping? What about the threshold for the wheel slipping?

THANK YOU. This helped me find the answer!

 

[haruspex]

THANK YOU. This helped me find the answer!

Glad you got the answer, but for didactic purposes here are the answers to some of the questions I asked:
Scenario 1: the cylinder slips at both contacts;
Scenario 2: the block slips at both contacts;
Scenario 3: the block slips at contact with cylinder and tips away from it.
All three are possible. Which happens depends on details of frictional coefficients and geometric ratios. The only way to be sure is to analyse each.
You did not explain how you figured out which happens. Is that the process you went through?

in the scenario that A slips, B slips, and Concrete doesn't slip, is the location of normal force at C and A remain at the center of each object?

No. When a uniform weight stands on a floor, the normal force is distributed across the contact, but on average passes through the mass centre of the object.
If you now apply a horizontal force, well before there is any tipping, the distribution of the normal forces changes such that the line of action of the average normal force shifts away from the applied force. That is necessary to preserve the torque balance.
When it reaches the "convex hull" of the contact area tipping can begin.

 

[lc99]

Glad you got the answer, but for didactic purposes here are the answers to some of the questions I asked:
Scenario 1: the cylinder slips at both contacts;
Scenario 2: the block slips at both contacts;
Scenario 3: the block slips at contact with cylinder and tips away from it.
All three are possible. Which happens depends on details of frictional coefficients and geometric ratios. The only way to be sure is to analyse each.
You did not explain how you figured out which happens. Is that the process you went through?No. When a uniform weight stands on a floor, the normal force is distributed across the contact, but on average passes through the mass centre of the object.
If you now apply a horizontal force, well before there is any tipping, the distribution of the normal forces changes such that the line of action of the average normal force shifts away from the applied force. That is necessary to preserve the torque balance.
When it reaches the "convex hull" of the contact area tipping can begin.

I think i took a slightly different approach. I solved this by finding the threshold for when the concrete block slips or tips. I had a set of equations for this scenario and figured out the the concrete block would tip before it slip.

I then looked at the wheel. I noticed that A and B slips before the concrete would be able to tip. Since this scenario 'works' , I solved for M. And the M i got was the answer.

 

[haruspex]

the concrete block would tip before it slip.

Yes, surprisingly that is true.
Well done.

 

[lc99]

Glad you got the answer, but for didactic purposes here are the answers to some of the questions I asked:
Scenario 1: the cylinder slips at both contacts;
Scenario 2: the block slips at both contacts;
Scenario 3: the block slips at contact with cylinder and tips away from it.
All three are possible. Which happens depends on details of frictional coefficients and geometric ratios. The only way to be sure is to analyse each.
You did not explain how you figured out which happens. Is that the process you went through?No. When a uniform weight stands on a floor, the normal force is distributed across the contact, but on average passes through the mass centre of the object.
If you now apply a horizontal force, well before there is any tipping, the distribution of the normal forces changes such that the line of action of the average normal force shifts away from the applied force. That is necessary to preserve the torque balance.
When it reaches the "convex hull" of the contact area tipping can begin.

i have a question about this. In lecture, we were told to prove that x (distance of normal force from the center of mass) is. in some scenerios, x has to be before the edge of the object for assumption that slipping occurs.

however, in this problem, i just assumed normal forces acts at the center. line of action. this assumption lessen the unknowns so i don't have to solve for x.

i don't understand when i should solve x?

 

[kuruman]

i don't understand when i should solve x?

Just before the block tips, x is at the edge of the block as you said. That sets how much horizontal force needs to be exerted on the block to tip it. If x is not right at the edge, you don't have to solve for it unless you need its value to find some other quantity or the question asks you specifically to find what it is.

 

[lc99]

Just before the block tips, x is at the edge of the block as you said. That sets how much horizontal force needs to be exerted on the block to tip it. If x is not right at the edge, you don't have to solve for it unless you need its value to find some other quantity or the question asks you specifically to find what it is.

I think i understand that tipping assumption, but this is what it says in my lecture

 

[kuruman]

There is no conflict with what your lecture notes say. I think of it slightly differently. The minimum force for slipping is given by $P_{slip}=\mu_s W$. The minimum force for tipping is $P_{tip}=W b/(2h)$. One can find values for each one of them. Whichever is smaller, that's what happens first. In this problem, I calculated the normal force $N_B$ when the wheel is just about to spin in place and compared its value to $P_{tip}$ and $P_{slip}$. Knowing $N_B$, one can find $x$ as an aside, but $x$ is not needed to figure out what happens first.

 

[lc99]

There is no conflict with what your lecture notes say. I think of it slightly differently. The minimum force for slipping is given by $P_{slip}=\mu_s W$. The minimum force for tipping is $P_{tip}=W b/(2h)$. One can find values for each one of them. Whichever is smaller, that's what happens first. In this problem, I calculated the normal force $N_B$ when the wheel is just about to spin in place and compared its value to $P_{tip}$ and $P_{slip}$. Knowing $N_B$, one can find $x$ as an aside, but $x$ is not needed to figure out what happens first.

It's just that when it comes to exams, they are give us points for showing ALL work. and not skipping steps. Like, even if we have the answer correct, how we got to it might be wrong

So, I'm like wondering if the idea behind checking if 0<= x<= b/2 is a valid proof for the slipping assumption

Maybe in this problem, i need to prove that the concrete block doesn't tip then i need to prove that x condition?

 

[kuruman]

It's just that when it comes to exams, they are give us points for showing ALL work. and not skipping steps. Like, even if we have the answer correct, how we got to it might be wrong

So, I'm like wondering if the idea behind checking if 0<= x<= b/2 is a valid proof for the slipping assumption

Maybe in this problem, i need to prove that the concrete block doesn't tip then i need to prove that x condition?

Your notes give you two ways for figuring out which occurs first and I gave you a third. All of them are correct. Be sure to understand the logic behind them - I don't think you're there yet - and then pick one that fits your thinking best. Here is a summary of the logic
Your notes method 1
Assume slipping occurs. When the block slips due to application of force P, $x$ must have moved closer to the edge. You know F is μ_sN, so find $x$. If $x<b/2$ then slip before tip; if $x>b/2$, then tip before slip.
Your notes method 2
Assume tipping occurs. Then $x=b/2$. Find the pushing force $P$ which is the same as the force of friction $F$. If $F<\mu_sN$, then tip before slip; if $F>\mu_sN$ then slip before tip.
My method
Assume that slipping occurs and find the pushing force $P_{slip}$ that is necessary to do that. This is also done in "Your notes method 1".
Assume that tipping occurs and find the tipping force $P_{tip}$ that is necessary to do that. This is also done in "Your notes method 2" and assumes $x=b/2$.
If $P_{tip} < P_{slip}$, then tip before slip; if $P_{tip} > P_{slip}$, then slip before tip.

 

[haruspex]

wondering if the idea behind checking if 0<= x<= b/2 is a valid proof for the slipping assumption

Imagine gradually increasing the applied force B. The frictional force F and displacement x of the normal force from the ground vary as functions of B.
Each of F and x is subject to an upper bound, and you want to know which bound is violated first. Is it valid to find B when one function is at its bound and check whether the other is within bounds?
If we abstract this to arbitrary functions f, g of some parameter t then it will not be valid since the other function may have violated the bound for a lower value of t, but then come back within bounds. The key requirement for validity is that the functions are strictly monotonic.
E.g. sketch graphs of F=F(B) and x=x(B). They need not be linear, but it is reasonably obvious that if B increases both F and x will increase. So if x is within bounds at two different values of B then it must be within bounds for all intervening values of B.