# Effects of friction on a system of bocks

Okay so I ran into this question while solving problems from my physics textbook and I need some help to understand the concept involved in it.

Question: In the shown arrangement if $f_1$, $f_2$ and $T$ be the frictional forces on $2$ kg block, $3$ kg block & tension in the string respectively , then what are their values?

My try: I tried to solve the problems by drawing the free body diagrams of the blocks as -

Doubt: But what I don't get is that how should I determine the direction of friction on the two blocks? Also, how do I figure out if the system is at rest or is it moving with some acceleration? The answer is certainly coming different in the two cases. Another doubt that I have is how do I know if there is maximum value of friction acting on the blocks or not and how do I determine that?

Please let me know how you would go about solving similar problems and what's the idea behind this friction thing.

To find the direction of the frictional forces $f_1, f_2$ you need to know the direction of motion of the blocks or, if they are at rest, the direction of the net force (excluding friction) acting—across the frictional surface—on the system of blocks.

Because we are not told otherwise, we will assume the blocks start at rest relative to the surface, and so we will find the direction of the net force acting on them (before friction) by simply adding you two applied force vectors:

$$\Sigma F = \vec{F_1} + \vec{F_2} = 1\text{N [Left]} + 8\text{N [Right]} = 7\text{N [Right]}$$

Given that we now know the direction of the net force acting on the blocks before gravity to be $\text{[Right]}$, we can say that the direction of the frictional forces are $\text{[Left]}$.

You already calculated the magnitude of the frictional forces in your attempt, and now we can add the direction to them:

$$f_1 = 2\text{N [Left]}$$ $$f_2 = 6\text{N [Left]}$$

And now, to calculate the tension, take the magnitude of the difference between the net forces on the two blocks:

$$T = |\Sigma F_2 - \Sigma F_1| = \left|\left(\vec{F_2} + \vec{f_2}\right) - \left(\vec{F_1} + \vec{f_1}\right)\right|$$ $$T = \left|2\text{N [Right]} - 3\text{N [Left]}\right| = 5\text{N}$$

I believe I (indirectly) answered all of your questions here, but feel free to ask if I didn’t.

First, do a trick and consider the motion of the center of mass. In this case you may forget about "inner" forces (tension in this case) and only consider external forces. The net force applied to a system is $7 N$ (without friction) and maximum total friction is $8 N$, so you can be sure that the system is at rest. Work your way up from that point.

Feel free to comment in case you get further difficulties. I will expand my answer then.

• quite impressive thought process I must say ! – Tanuj Nov 7 '17 at 14:43