Why Does a Wheel Move and How Does Torque Influence Its Dynamics?

[erensatik]

Homework Statement

The truss in the figure is made of light aluminum struts freely pivoted at each end. At C is a roller which rolls on a smooth plate. When a workman heats up member AB with a welding torch, it is observed to increase in length by an amount x, and the load W is thereby moved vertically an amount y.
a) Is the motion of W upward or downward?
b) What is the force in the member AB (including the sense, i.e., tension or compression).

Relevant Equations

No relevant equation

Why the wheel moves? I intuitively think that it should move to right but I don't really understand why. When AB gets longer by x each end gets longer by x/2 so W goes to right and down. This will create torque and body will move. This explanation seems valid but I can't calculate anything and you should do it using energy principles.

 

[haruspex]

Each triangle of three struts not containing AB is fixed in shape. Can you reduce the system to four rigid objects? Then ask which way the flag moves and hence which way C moves.

 

[erensatik]

Each triangle of three struts not containing AB is fixed in shape. Can you reduce the system to four rigid objects? Then ask which way the flag moves and hence which way C moves.

I thought the system is like 4 rod all connected to AB. I don't know this is valid and even if I assume it's true I can't calculate force on AB. Can you make it more clear?

 

[haruspex]

I thought the system is like 4 rod all connected to AB.

Yes. Call the joint at the top with the flag F and the other joint adjacent to A and B joint E. If AB lengthens, what happens to angle EBF?

 

[erensatik]

Yes. Call the joint at the top with the flag F and the other joint adjacent to A and B joint E. If AB lengthens, what happens to angle EBF?

EBF gets smaller I guess. But I can't solve it. Can you solve it completely?

 

[jbriggs444]

EBF gets smaller I guess. But I can't solve it. Can you solve it completely?

So as angle EBF gets smaller, what happens to the left hand structure that contains segment EB? Does it change its lean angle? Which way?

Similarly, what happens to the right hand structure that contains segment BF? Does it change its lean angle? Which way?

Does W move vertically up or down as a result?

 

[Lnewqban]

I thought the system is like 4 rod all connected to AB. I don't know this is valid and even if I assume it's true I can't calculate force on AB. Can you make it more clear?

Welcome, erensatic!

I don't see how this problem is related to conservation of energy, other than any geometric change to this armature changes the potential energy of mass W (aluminum struts are light enough as to the weight of the armature not to be considered).

I believe that you don't need to calculate the force on AB, only to conclude the type of load that that aluminum struts is supporting, compression or tension.

If you were to grab points A and B with each of your hands, and then somebody completely removes the AB aluminum strut, what the action of gravity on mass W would force you to do to keep the geometry of the armature as is?

Intuitively, do you believe that your arms would need to restrain points A and B from moving away from each other (hence, AB strut has been under tension load) or to move toward each other (hence, strut has been under compression load)?

You can consider the red and blue portions of the armature as rigid planes.
In that case, and with AB strut removed, the blue plane is free to rotate around the left grounded pivot, while the red plane is free to simultaneously rotate around sliding pivot C.
At the same time, A-Red-plane and A-Blue-plane struts are both free to rotate about their respective pivots.

 

[haruspex]

EBF gets smaller I guess. But I can't solve it. Can you solve it completely?

Model it with two index fingers. Make fists at eye level, facing each other, index fingers extended. Put the tip of the left one half way along the right one (point B). Keep the base of the left finger in a fixed position, and only allow the base of the right to move horizontally.
Which way do you have to move your right hand to reduce the angle the left finger makes to the top half of the right? Does it require moving the fists closer or further apart?
Does B move up, down, or only horizontally?

 

[erensatik]

Thank you all for replying. I think now I can imagine what's going on. A and B would try to move away if you remove AB strut so there is tension and B goes down. I get these. But part b is still a problem. Because answer that the book gives is W(y/x) which is not just tension or compression. I have no idea how to calculate it. Please help.

 

[haruspex]

Thank you all for replying. I think now I can imagine what's going on. A and B would try to move away if you remove AB strut so there is tension and B goes down. I get these. But part b is still a problem. Because answer that the book gives is W(y/x) which is not just tension or compression. I have no idea how to calculate it. Please help.

My immediate response was:
"Use conservation of work. If the tension in the strut is T, how much work is need to grow its by length x?"
But I have a doubt about this now. I'll take a closer look later.

Later: ... it's ok, I believe.

 

[jbriggs444]

My immediate response was:
"Use conservation of work. If the tension in the strut is T, much work is need to grow its by length x?"
But I have a doubt about this now. I'll take a closer look later.

For small displacements, that matches the strategy I had in mind.

 

[Lnewqban]

... But part b is still a problem. Because answer that the book gives is W(y/x) which is not just tension or compression. I have no idea how to calculate it. Please help.

For any ideal mechanism, input energy (work in) = output energy (work out).
... and work = force times distance.

 

[erensatik]

Oh god, I was thinking y as a horizontal movement of W. I kept drawing triangle and used pythagoras for an hour. It's clear now.
I thank everyone for helping me out. By the way, this forum is awesome! Liked it.