- #1
Buffu
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Homework Statement
Find the accelaration of ##M_1## in the given system if ##F = 0##.
Homework Equations
The Attempt at a Solution
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##x_3 -x_1 = k \iff \ddot x_3 = \ddot x_1##
and ##h - y_3 + x_3 - x_2 = l \iff \ddot y_3 + \ddot x_2 = \ddot x_3 \qquad (*)##
h is the height of block ##M_1## and ## l ## is the length of string between ##M_2## and ##M_3##.
Now from the Free body diagram,
##-M\ddot x_1 = N^{\prime \prime \prime}##
##T = M_2 \ddot x_2##
##N^{\prime \prime \prime} = -M_3\ddot x_3##
##\therefore -M\ddot x_1 = M_2 \ddot x_2 - M_3\ddot x_3 \qquad (1)##
Now from vertical force on ##M_3##,
##M_3 - T = -M_3 \ddot y_3##
##-M_3 g + M_2 \ddot x_2 = M_3 \ddot y_3##
Substituting for ##\ddot y_3## in ##(*)##
##\ddot x_3 = x_2 + \dfrac{-M_3 g + M_2 \ddot x_2 }{M_3}##
Solving for ##\ddot x_2##
##x_2 = \dfrac {M_3(\ddot x_3 + g) }{M_3 + M_2}##
Substituting this in ##(1)##
##-M_1 \ddot x_1 = \dfrac {M_3M_2(\ddot x_3 + g) }{M_3 + M_2} - M_3\ddot x_3##
Since ##\ddot x_1 = \ddot x_3##
##-M_1 \ddot x_1 = \dfrac {M_3M_2(\ddot x_1 + g) }{M_3 + M_2} - M_3\ddot x_1##
Solving for ##\ddot x_1##
##\ddot x_1 = \dfrac{-g(M_2M_3)}{M_1M_2 + M_3M_1 - M_3^2}##
Which is incorrect as the given answer is ##\ddot x_1 = \dfrac{-g(M_2M_3)}{M_1M_2 + M_3M_1 \color{red}{ + 2M_2M_3 +} M_3^2}##.
What is the problem ?