Given the setup shown in Fig. Block A, B, and C have masses $$m_A=M$$ and $$m_B= m_C=m$$. The strings are assumed massless and unstretchable, and the pulleys frictionless. There is no friction between blocks B and the support table, but there is friction between blocks B and C, denoted by a given coefficient $$\mu$$.
a. In terms of the given, find (i) the acceleration of block A, and (ii) the tension in the string connecting A and B.
b. Suppose the system is related from rest with block C. near the right end of block B as shown in the above figure. If the length L of block B is given, what is the speed of block C as it reaches the lift end of block B? Treat the size of C as small.
c. If the mass of block A is less than some critical value, the blocks will not accelerate when released from rest. Write down an expression for that critical mass. 


Apply constraint equation on strings, the length of strings is constant. Differentiate twice to get relation between of acceleration of block A, B, and C be a, b, and c, respectively.
 $$l_1 +l_2$$ = constant
and $$l_3 + l_4 $$ constant
$$l_1+ l_2 = 0 \Rightarrow |b| = |c|$$
$$l_3 + l_4 = 0 \Rightarrow |a| = |b|$$
From which we get a=b=c.
From FBDs Of A, B, and C   [Fig. (a)],
Writing equations of motion for block A:
$$mg - T = Ma$$                        (i)
For block B, $$T - T_1 -\mu mg = ma$$              (ii)
For block C, $$T- \mu mg = ma $$                       (iii)
Solving equations (i), (ii) and (iii), we get
$$a= (\dfrac{m-2\mu m}{m+2m})g$$                       (iv)
Putting a in Eq. (i) , we get 
$$ mg -T = M (\frac{M-2\mu m}{M+2m}) g \Rightarrow T = \frac{2mMg(1+\mu)}{(M+2m)}$$
b. As there is relative motion between blocks, we apply 
$$V_{rel}^2 = V_{rel}^2 +2a_{rel} S_{rel}$$
If system is released from rest, $$u_{rel} =0$$
$$v_{rel}^2 = 2a_{rel} S_{rel} \Rightarrow v_{rel} =\sqrt{2a_{rel} S_{rel}}$$
$$a_{rel} = 2a $$ and $$S_{rel} L$$
$$\Rightarrow v=\sqrt{\dfrac{4gL(M-2\mu m)}{(M+2m)}}$$
c. If blocks will not accelerate, then put a=0 in Eq. (iv) to get $$M=2\mu m$$.



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