Trouble interpreting fictitious forces for block on wedge problem

[shadowplay]

Homework Statement

This is a standard block on wedge problem - we have an incline of angle $$\alpha$$, a block of mass m on wedge of mass M. The block is released from the inclined surface. The wedge is not fixed and can accelerate. The question is typical - find the horizontal accelerations of the block and wedge a₁ and a₂ respectively, and the normal foces N₁ and N₂ (in terms of the masses the angles and g and stuff). I know this is a standard problem and I in fact know how to solve it using the constraint invoking the geometry of the incline.

However, I really want to do this problem using fictious forces. I want to use a reference frame in which the wedge is fixed.

Homework Equations

Let inertial frame be S, accelerated one be S' (not rotated).

m_a = F + F_fictitious

The Attempt at a Solution

I thought the above equation wold make this problem fall out very quickly using the above equation, because the acceleration of the wedge with respect to the inertial frame (and hence the fictitious force) is easy to calculate once we know the normal force N₁.

Now, in the frame fixed with respect to the wedge, I thought we would have N₁=mgcos$$\alpha$$ (and consequently an acceleration of the block down the incline of gsin$$\alpha$$). But this isn't the case, right? Why not, though? And how would we solve for N₁ using our fictitious force framework?

 

[ashishsinghal]

Now, in the frame fixed with respect to the wedge, I thought we would have N₁=mgcos$$\alpha$$ (and consequently an acceleration of the block down the incline of gsin$$\alpha$$). But this isn't the case, right? Why not, though? And how would we solve for N₁ using our fictitious force framework?

It is not so because the fictitious force is also acting. So you should take that into account as well. Show complete working preferably with a diagram so we could tell where you are wrong.

 

[shadowplay]

Well I haven't been able to use fictitious forces, so there is no point in me writing down the working out. Fictitious forces act IN addition to the forces present in inertial frames, like N1. Shouldn't N1 have the same value in the inertial frame and the frame in which the block is fixed? So how do we go about finding N1 (without solving the problem entirely in the inertial frame, which defeats the entire point/advantage of using the noninertial frame of reference)?

 

[Doc Al]

Shouldn't N1 have the same value in the inertial frame and the frame in which the block is fixed?

It does; the value of a 'real' force such as N does not depend on the reference frame. But that value is different from what it would be if the wedge were fixed. (Which is an entirely different problem.)

So how do we go about finding N1 (without solving the problem entirely in the inertial frame, which defeats the entire point/advantage of using the noninertial frame of reference)?

Call the acceleration of the wedge 'a'. Then write your force equations for the block in the accelerating frame of the wedge. (How is the block's motion constrained?)

 

[rwisz]

As a scientist, it is important to approach problems from different perspectives and explore all possible solutions. Using fictitious forces to solve this standard block on wedge problem is a valid approach, but it may not provide a clear understanding of the underlying physics. It is important to remember that fictitious forces are not actual forces, but rather mathematical constructs used to simplify calculations in non-inertial reference frames. In this case, using fictitious forces may lead to confusion or difficulty in interpreting the results.

In order to solve for the normal force N1 using fictitious forces, we would need to consider the acceleration of the wedge in the inertial frame. This would involve taking into account the acceleration of the wedge due to the block's weight and the normal force N2 exerted by the wedge on the block. This can become quite complex and may not provide a clear understanding of the forces at play.

In contrast, using the constraint invoking the geometry of the incline approach provides a more direct understanding of the forces involved and their relationships. This approach takes into account the weight of the block and the normal force N1 exerted by the incline on the block, without the need for considering fictitious forces.

In conclusion, while it is important to explore different approaches to problem-solving, it is also important to consider the most effective and intuitive method for understanding the underlying physics. In this case, the constraint invoking the geometry of the incline approach may be more useful in interpreting the forces involved in the block on wedge problem.