Question

three masses m1 ,m2,and m3 are attached to a string pulley .all three masses are held at rest and then released . if m3 remains at rest ,then find relation in m1 ,m2 and m3

Answer 1

Answer:

Given data states that there are three masses $m_1, m_2\quad and\quad m_3$, all are tied to a string over pulley at rest and then released, therefore, we have an equation

$-T-{ m }_{ 1 }g\quad =\quad { m }_{ 1 }a\quad and\quad { m }_{ 2 }g-T\quad =\quad { m }_{ 2 }a.$

Add these equation and we will get the acceleration $a-{ m }_{ 2 }g-{ m }_{ 1 }g\quad =\quad ({ m }_{ 1 }+{ m }_{ 2 })\times a$

$\Rightarrow a\quad =\quad \frac { ({ m }_{ 2 }g-{ m }_{ 1 }g) }{ ({ m }_{ 1 }+{ m }_{ 2 }) }$

Substitute the value of a into the equation to find Tension T,

we get $-T\quad =\quad { m }_{ 1 }g+{ m }_{ 1 }\times a$

$\Rightarrow \frac { (2{ m }_{ 1 }{ m }_{ 2 }g) }{ ({ m }_{ 1 }+{ m }_{ 2 }) }$

The third mass is at rest, therefore tension will be

$-T\quad =\quad 2T\quad =\quad \frac { (4{ m }_{ 1 }{ m }_{ 2 }g) }{ ({ m }_{ 1 }+{ m }_{ 2 }) }$

. Therefore, as ${ m }_{ 3 }$ at rest,

${ m }_{ 3 }g\quad =\quad \frac { (4{ m }_{ 1 }{ m }_{ 2 }g) }{ { (m }_{ 1 }+{ m }_{ 2 }) }$Rearrange the given expression,  

we get – $-{ m }_{ 1 }{ m }_{ 3 }+{ m }_{ 2 }{ m }_{ 3 }\quad =\quad 4{ m }_{ 1 }{ m }_{ 2 }$