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Question Number 108922 by ajfour last updated on 20/Aug/20

Commented by ajfour last updated on 20/Aug/20

       Neglect friction. Find       initial acceleration of M.

$$\:\:\:\:\:\:\:{Neglect}\:{friction}.\:{Find} \\ $$$$\:\:\:\:\:{initial}\:{acceleration}\:{of}\:{M}. \\ $$

Answered by mr W last updated on 20/Aug/20

Commented by mr W last updated on 20/Aug/20

a_1 =A+a  T=m_0 a  T=MA  mg−T=ma_1 =m(A+a)  ⇒a=(M/m_0 )A  ⇒mg−MA=m(A+(M/m_0 )A)  ⇒A=(g/(1+(M/m)+(M/m_0 )))  ⇒a=(g/(1+(m_0 /m)+(m_0 /M)))

$${a}_{\mathrm{1}} ={A}+{a} \\ $$$${T}={m}_{\mathrm{0}} {a} \\ $$$${T}={MA} \\ $$$${mg}−{T}={ma}_{\mathrm{1}} ={m}\left({A}+{a}\right) \\ $$$$\Rightarrow{a}=\frac{{M}}{{m}_{\mathrm{0}} }{A} \\ $$$$\Rightarrow{mg}−{MA}={m}\left({A}+\frac{{M}}{{m}_{\mathrm{0}} }{A}\right) \\ $$$$\Rightarrow{A}=\frac{{g}}{\mathrm{1}+\frac{{M}}{{m}}+\frac{{M}}{{m}_{\mathrm{0}} }} \\ $$$$\Rightarrow{a}=\frac{{g}}{\mathrm{1}+\frac{{m}_{\mathrm{0}} }{{m}}+\frac{{m}_{\mathrm{0}} }{{M}}} \\ $$

Commented by ajfour last updated on 21/Aug/20

Straight way & great way, thank  you Sir! Can we discuss what   happens after a while when m  even accelerates towards left..?

$${Straight}\:{way}\:\&\:{great}\:{way},\:{thank} \\ $$$${you}\:{Sir}!\:{Can}\:{we}\:{discuss}\:{what}\: \\ $$$${happens}\:{after}\:{a}\:{while}\:{when}\:{m} \\ $$$${even}\:{accelerates}\:{towards}\:{left}..? \\ $$

Commented by mr W last updated on 21/Aug/20

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