# NLM Problem: More variables than equations

• Seeker220
In summary, the conversation discusses the attempt at solving a problem involving two blocks, ##m_1## and ##m_2##, connected by a string and placed on an inclined plane. The chosen reference frame is ##m_2## and a free body diagram is drawn. Equations are obtained by choosing axes along and perpendicular to acceleration of ##m_1## with respect to ##m_2## (##a'##). However, there are discrepancies in the FBD and the equations, leading to a discussion on the net horizontal force and acceleration of the two blocks as a system. The conversation ends with the suggestion to present accurate FBDs of each block.

#### Seeker220

Homework Statement
Two blocks initially at rest are kept in contact as shown in the figure. A force F is applied on the left block. The blocks are of masses ##m_1## and ##m_2##. Find the acceleration of the blocks. All surfaces are smooth.
Relevant Equations
##Force = Mass \times Acceleration##

Attempt at Solution::

##m_2## is chosen as reference frame and FBD is drawn as shown above. We get the following equations:

From ##m_1## by choosing axes along and perpendicular to acceleration of ##m_1## w.r.t ##m_2## (##a'##):

$$m_1 a' + m_1 a \cos{\theta} + m_1 g \sin{\theta} = N_1 \sin{\theta} + F \cos{\theta}$$
$$N+N_1 \cos{\theta} + m_1 a \sin{\theta} = m_1 g \cos{\theta} + F \sin{\theta}$$

From ##m_2## by choosing horizontal and vertical axes:
$$N_2 = N \cos{\theta} + m_2 g$$
$$m_2 a = N \sin{\theta}$$

Which gives us four equations, but five variables, namely ##a',a,N_1 ,N_2, N##

Your FBD is incorrect. ##m_1a## and ##m_2a## are not forces. If you consider the two blocks together as your system, what is the net horizontal force acting on that system? What is its horizontal acceleration? Start from there then draw two separate FBDs for each block.

topsquark, erobz and Seeker220
Please label the coordinates on the diagram, and the associated coordinate with each equation you have written down.

topsquark and Seeker220
kuruman said:
Your FBD is incorrect. ##m_1a## and ##m_2a## are not forces. If you consider the two blocks together as your system, what is the net horizontal force acting on that system? What is its horizontal acceleration? Start from there then draw two separate FBDs for each block.
umm.. ##m_1a## and ##m_2a## are pseudo forces because accelerating frame ##m_2## (with acceleration a towards right) is my Reference frame.

Net horizontal force on the two blocks as a system is F.
Sum of mass × horizontal acceleration of two blocks is F.

But how can you comment about their individual horizontal acceleration?

Think about a' and N1. can both be nonzero?

topsquark, erobz and Seeker220
Why ##N_1## participating in the sum of the forces if you assume the small mass is accelerating up the incline?

There are probably a few errors here. You should do as @kuruman suggest and present accurate FBD's of each block.

topsquark and Seeker220
haruspex said:
Think about a' and N1. can both be nonzero?
I think this actually solves it. Ok trying and editing to final solution...

So what did you get for ##a## and ##a'## as labeled in your diagram?

Seeker220 said:
I think this actually solves it. Ok trying and editing to final solution...
Please post the edited solution in a new post in this thread.
That's preferrable to editing Post #1 itself, which gets very confusing for anybody reading the thread at a later time.

kuruman, erobz and topsquark