Physics problem

[Encrypted]

Hey guys, I'm struggling a little bit with this physics problem (look at the attachment for a diagram of the situation about to outline).

 

There is a cart, upon which a block of a material of uniform density, with width w and height h rests. The coefficient of friction between the block and the cart is enough so that when the cart begins to accelerate, the block cannot slide, but instead topples over.

 

At what acceleration will the block topple? I need to find a general solution for this situation.

 

These are the forces I can identify:

 

a) the weight of the block, mg (acting through the center of mass, which lays in the geometric center of the object, since it is of uniform density)

b) the normal force to the weight of the block (acting through the center of mass, in the direction opposite to that of mg)

c) the force of friction between the block and cart (this is the force that is accelerating the cart, and it acts along the surface of the block)

 

The problem is, I can't figure out which force causes the counterclockwise torque that causes the block to topple over, and I know for a fact it's not one of the above three forces. I know it's a counterclockwise torque because I did some modelling using a sheet of paper and an eraser, which acted as my block). Our teacher says that the force responsible for the torque would be the "pseudo force", ma, which acts in the direction parallel to the acceleration, a, and through the center of mass.

 

That doesn't make sense either, because that force, ma, would cause a torque in the clockwise direction, not counterclockwise.

 

WTH is happening?

[swansont]

Forces acting through the center of mass can't exert a torque. So, what's left? And why don't you think it can exert the necessary torque?

[Encrypted]

Well, because I decided the best place (and the most realistic place) for the fulcrum would be the lower left corner of the block. In that case, none of the forces would be able to create a torque, because the force of friction passes through the fulcrum.

 

Swansont, can you get on SFN IRC Chat? Then we can discuss this much easier in real time.

[theCPE]

the acceleration of the cart the block rests on creates a pseudo force that acts in the opposing direction of the carts acceleration at the top of the block.

[Mr Skeptic]

The pseudoforce you are looking for is inertia. I trust that with that you can figure the rest out, or do you need more help?

[Encrypted]

That makes some sense, that the pseudo force would be inertia. But where would it be located (or where would its "line of action" be)?

[swansont]

Well, because I decided the best place (and the most realistic place) for the fulcrum would be the lower left corner of the block. In that case, none of the forces would be able to create a torque, because the force of friction passes through the fulcrum.

 

Swansont, can you get on SFN IRC Chat? Then we can discuss this much easier in real time.

 

I should amend what I said to "can't cause a rotation about the center of mass" because it's wrong by itself.

 

Anyway, if you want the rotation point to be the corner of the block, you have to put yourself in the accelerating frame, and then you have the pseudoforce that you and others have mentioned.

 

But I think that friction acting to (initially) cause a rotation about the center of mass gives the exact same answer.

[theCPE]

That makes some sense, that the pseudo force would be inertia. But where would it be located (or where would its "line of action" be)?

 

It acts at the top of the block opposite the acceleration of the cart.

 

It produces the counterclockwise rotation you are looking for.

 

There are two ways to view it.

 

Inertia or as a force.

 

If you view it as inertia, think, what would happen if a block was standing on something that suddenly accelerated. Which way would the block rotate.

 

If you view it as a force then the block is a rigid body. The acceleration of the cart (and it not slipping due to high coefficient of friction) is a force at the bottom of the block in the direction of the acceleration (frictional force). The rigid body translates that to a force at the top of the block in the opposing direction.

 

Put a ruler or pencil or something on a table. Push perpendicular to it on one end. It rotates. There isn't a distinguishable difference to create that rotation whether you pushed in one direction at one end or the opposite direction at the other end. In this case the frictional force thanks to the coefficient of friction between the cart and the block is your finger.

 

Maybe that makes sense.

[Encrypted]

Yup, I solved the problem, and it makes much more sense to me now. My teacher had made a number of critical errors (or used unnecessary steps), such as combining the force of friction and the normal force into another force which he labeled "Fr" (subscript r). He also put down the pseudo force ma in the wrong direction.

 

Anyways, somehow, he still ended up with the right answer, which is either very wierd, or very lucky.