Can anyone else solve this? Inclined plane super problem

[carrotstien]

Ok, maybe nothing is super about this, but I created this problem when studying for a mechanics test (first year of college) and none of my friends can do it. I gave the problem to my professor, and never got a reply, hmm...

 

Anyway, I am just really interested to see how many people could actually solve it besides me. Just a challenge.

 

You have an massless frictionless inclined plane that is attached to a frictionless rail so that it can only move left and right - no rotation or vertical movement. It has an angle of theta, where theta isnt 0 and is equal to or less than 90. On this incline is a block of mass M. You apply a force of F on the vertical part of the incline so as to make it move along the direction of the rail (so the the force is acting parallel to the rail).

There is no gravity or friction anywhere in the problem.

 

What is the acceleration of the incline?

 

If you could solve this, then solve the same situation but with friction coefficients, static and kinetic (so you have to solve for the set up that would make the block slip and the set up that wouldnt), and a mass of the incline N. Still no gravity, or friction at the rail.

 

Good luck!

[tvp45]

I'll only solve the first part for now. Maybe later, if I'm satisfied by your response, I'll do the second.

 

The plane is not a material object. It is a 2d ideal. Force cannot act on it to produce a change in momentum; on the other hand, it has no mass (and thus no inertia) to resist a force. Thus, we would find ourselves in the strange position of finding dv/dt = 0/0 and, in the real world, that is undefinable. Keep in mind that we are not taking a limit as the mass approaches 0; you specified a massless plane.

 

Thus, the closest answer is that it does not accelerate. The block M simply passes through it since there is nothing to prevent that. It is really quite close to the invisible plane at the goal line in the NFL; no player can ever hit it so it doesn't move.

[Klaynos]

I think, I solved something identicle to that the other day, it's trivial if you use legrangian dynamics.

 

I'll work through it tomorrow if I'm feeling less ill!

[carrotstien]

First of all, you cant have a massless object in reality whose shape is an incline. But this isnt reality, its an ideal situation. Maybe I should include that the incline, even though is massless, does not permit other object to go through it, just like normal objects have surface normal forces when pushed by other objects.

If you want to more realistic problem, imagine the incline having a mass, and do the same problem. You will get a solution, that when you set the variable of the mass of the incline to be 0, you get the solution to my original problem.

[tvp45]

No, I won't work the second half. I'm not satisfied with your response. Physically, you cannot set the mass to zero and have a force act on this plane; you can only do the limit as the thickness becomes very small.

[carrotstien]

What are you talking about? Why cant I have a massless object apply a force - ropes and pullies apply forces even when they are considered massless. Like I said, if it bothers some weird part of you that this problem isnt very realistic one way or another, give both things mass and volume, add wind resistence - the more stuff you add, the more complicated the problem gets. Trying doing to simpler way first, if you cant, ill understand.

[swansont]

No, I won't work the second half. I'm not satisfied with your response. Physically, you cannot set the mass to zero and have a force act on this plane; you can only do the limit as the thickness becomes very small.

 

If the plane were moving on its own this would be true, since you could not conserve momentum (i.e. in the limit of F going to zero). But since the object on the incline has mass, this is merely the approximation that the mass of the incline is small in comparison.

[tvp45]

If the plane were moving on its own this would be true, since you could not conserve momentum (i.e. in the limit of F going to zero). But since the object on the incline has mass, this is merely the approximation that the mass of the incline is small in comparison.

So you have have to decide whether the plane is massless or whether the mass is very small compared to the object on the incline. And, no, they are not the same by any means. But, my answer remains the same: I don't do trick questions that lack physical meaning.

[thedarkshade]

Massless and frictionless are two adjectives which are unknown for realistic situations so this one is completely idealistic, so we'll treat it that way. As you say that this plane is incline and frictionless, any forced acting on it (vertically as you said) would just slip (because of frictionless situation) and it would also slip from the rail too (as it is also frictionless), so no force (acting vertically) would be able to make this plane move. This slip is caused from the frictionless situation and also because of the angle which cannot be zero. If the force acts from left, than it would make the plane move infinitely with a constant velocity.

At least I picture it that way!

[swansont]

So you have have to decide whether the plane is massless or whether the mass is very small compared to the object on the incline. And, no, they are not the same by any means. But, my answer remains the same: I don't do trick questions that lack physical meaning.

 

No, from a solution perspective they are identical within the constraints I outlined; you set the term to zero. And I guess you've never taken an advanced physics course. Trick* questions without physical meaning tend to crop quite often.

 

*This doesn't qualify according to my definition of a trick question. In a real trick question, the knowledge that invalidates the solution method isn't found in the problem, nor is it an expected piece of knowledge for the type of problem, e.g. asking how far a muon would travel if it went at speed v for time t, where t is very much longer than its lifetime, and then saying that the d=vt answer is wrong because the particle would have decayed. That is a trick question. "The elephant whose mass may be ignored" is standard physics fare and really shouldn't cause much consternation.

[tvp45]

No, from a solution perspective they are identical within the constraints I outlined; you set the term to zero. And I guess you've never taken an advanced physics course. Trick* questions without physical meaning tend to crop quite often.

 

*This doesn't qualify according to my definition of a trick question. In a real trick question, the knowledge that invalidates the solution method isn't found in the problem, nor is it an expected piece of knowledge for the type of problem, e.g. asking how far a muon would travel if it went at speed v for time t, where t is very much longer than its lifetime, and then saying that the d=vt answer is wrong because the particle would have decayed. That is a trick question. "The elephant whose mass may be ignored" is standard physics fare and really shouldn't cause much consternation.

 

No, I didn't set the term to zero. The OP stated a massless, frictionless plane, not an incline, not a very thin incline, not something within a limit. I'm OK with that, except then he wants to say "well, it's approximately massless." I'm OK with that also, but I'm not going there.

 

I did my time in solving problems without physical meaning. Now I don't have to. And, I am perfectly comfortable with assuming a cow is a sphere, provided halfway through, you don't say "what about the legs".

 

Let the OP provide his answer.

[carrotstien]

I really dont see what the problem is. If the question isnt perfectly stated, fine, im sorry - but that doesnt mean that you should, perfectly understanding the situation, choose to avoid answering. This remind of my physics professor. I gave him this problem, before I included the rails under the incline - and instead of solving it, he showed me a screenshot of his physics simulator showing that the wedge tips over. Sheesh, thats obviously not what Im going for.

 

Now, the solution for people that care about it, and dont spend time writing paragraphs of excuses for not answer the question at hand.

 

Every force in the universe has to exist with a reaction force. That means, that the horizontal force applied onto the incline must equal the total reverse horizontal force from the system. In this case, the only source of this force is the blocks normal force upon the plane, which acts at an angle. So, if N is the normal force, Nsin@ = F. If E is the acceleration of the block (not the incline) than EMsin@ = F. E = F/(Msin@)

 

That deals with the physics of it, now the geometry. There is no friction - that means that the only way that the wedge could push the block is perpendicularly to the surface no matter the motion of the wedge. (See attatchment [edit, oops its in the next comment]). So, for every dx is an small movement of the wedge, and db is a small movement of the block, then dxSin@ = db. Taking the double time derivative gives u that the acceleration of the Block is Sin@ * the acceleartion of the incline.

 

So you are left with --> E = Asin@ = F/(Msin@)

 

So, A = F/(Msin@sin@)

 

Was that so hard that everyone had to make such a fuss over it?

I dont mean to sound mean at anyone who actually tried. The purpose of this was to test your understanding of physics and intuition, not to test you problem solving skills or your skills of higher level physics (for those of you who think this way, why didnt you include the bending of space time and all the mini black holes located in the Nth dimension?)

 

Interestingly enough, complicating the problem with more realistic stuff will get you answers that, when simplified by 0ing a variable, get you back to where to started.

 

If A is the mass of the incline, and B is the mass of the block then

Acceleration of incline = F/(A + Bsin@sin@)

 

If the above situation, but with friction between the blocks (and the blocks are slipping)

Acceleration of incline = F/(A + Bsin@sin@ + B[mu]cos@sin@)

 

If the above but with gravity

Accel = (F-GBcos@sin@ - [mu]GBcos@cos@)/(A + Bsin@sin@ + B[mu]cos@sin@)

 

If you even add friction to the rails

 

Accel = (F-GBcos@sin@ - [mu]GBcos@cos@ - GA[muRAIL] - GB[muRAIL]cos@cos@)/(A + Bsin@sin@ + B[mu]cos@sin@ + B[muRAIL]cos@sin@)

 

------------------

A final note, physics is the science which tries to predict and explain stuff that happens in nature. When doing physics problems, you cant expect to deal with real things, since we still dont know for sure, what exactly is going on everywhere, and we probably never will. Physics is a study where every law and the such is built up upon everything before. Not counting relativity, how come I have never heard anyone question why F = MA. This is a relationship gotten through common sense, intuition, and empirical data - however there is still no concrete reason why mass has inertia. So people, relax a little, and when shown a problem, solve it, then question its validity - or else you are nothing better than a good arguer.

[carrotstien]

attachments...are attached, lol