There are three blocks of masses m 1 = m , m 2 = 4 m and m 3 = m . Two block system is kept over a smooth hemisphere with two different arrangement one by one as shown in figure (i) & (ii) respe

Question

There are three blocks of masses m 1 ​ = m , m 2 ​ = 4 m and m 3 ​ = m . Two block system is kept over a smooth hemisphere with two different arrangement one by one as shown in figure (i) & (ii) respectively and are connected with massless & inextensible string. In arrangement figure (i), after releasing the system from rest acceleration of blocks is found to be ( a = 2 5 3 ​ ​ ) m / s 2 . In second arrangement block m 1 ​ & m 2 ​ are kept symmetrically. Find the acceleration of block (in m / s 2 ) after releasing the system from rest according to figure (ii).

TestHub · Accepted Answer

since blocks are connected, acceleration of both blocks is same and it is given as : $a = \frac{5 3}{2} m / s^{2}$ Net pulling force : $m_{3} g sin θ$

$m_{3} g sin θ = (m_{1} + m_{3}) a$

$m_{1} = m_{3} = m$

so, $mg sin θ = 2 m \times \frac{5 3}{2}$

$sin θ = \frac{3}{2} \Rightarrow θ = 6 0^{\circ}$

Now

Net pulling force :

$m_{2} g sin (\frac{θ}{2}) - m_{1} g sin (\frac{θ}{2})$

so $m_{2} g sin (\frac{θ}{2}) - m_{1} g sin (\frac{θ}{2}) = (m_{1} + m_{2}) a_{2}$

$m_{2} = 4 m & m_{1} = m$

$(4 m - m) g sin 30 = 5 ma_{2}$

$3 mg \times \frac{1}{2} = 5 ma_{2}$

$\Rightarrow a_{2} = 3 m / s^{2}$

so ans. is 3.

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