A body of mass M is hanging by an inextensible string of mass m. If the free end of the string accelerates up with constant acceleration a, find the variation of tension in the string as a function of the distance measured from the mass M (bottom of the string).

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Solution

Let T be the tension in the string at a distance x from its bottom. Referring to the FBD, we have the following equations:
Force equation for $m_{1}$ :
$T = m_{1} g - M g = (M + m_{1}) a$ .......(i)
Force equation for $m_{2}$ :
$T_{0} - m_{2} g - T = m_{2} a$ .......(ii)
Any one Eq. can be used
Substituting $m_{1} = \frac{m}{l} x$ in equation (i), we have
$T = (M + \frac{m x}{l}) (g + a)$
At $x = 1$ , $T = (M + m) (g + a) = F (s a y)$

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