Question

# 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.

Solution

## 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. $$l_1 +l_2$$ = constantand $$l_3 + l_4$$ 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 $$M=2\mu m$$.Physics

Suggest Corrections

0

Similar questions
View More

People also searched for
View More