Question

Find the tension in the string and acceleration of both blocks ?​

Answer 1

$\\ {\mathfrak{\underline{\purple{\:\:\: Given:-\:\:\:}}}}$

$\:\:\:\:\bullet\:\:\:\sf{ Angle \: of \: edge = \theta}$

$\:\:\:\:\bullet\:\:\:\sf{Mass \: of \: blocks = m_{1} \: and \: m_{2} }$

${\mathfrak{\underline{\purple{\:\:\:To \:Find:-\:\:\:}}}}$

$\:\:\:\:\bullet\:\:\:\sf{Tension \: in \: the \: string }$

$\:\:\:\:\bullet\:\:\:\sf{Acceleration \: of \: both \: blocks}$

${\mathfrak{\underline{\purple{\:\:\: Calculation:-\:\:\:}}}}$

☯ $\sf\underline{For \: mass \: of \: block( m_{1}) }$

$\dashrightarrow\:\: \sf{ m_{1}g - T= m_{1}a ..... (1) }$

☯ $\sf\underline{For \: mass \: of \: block( m_{2}) }$

$\dashrightarrow\:\: \sf{ \dfrac{T}{2} - m_{2}g \: sin \: \theta = \dfrac{m_{2}a}{2} }$

$\dashrightarrow\:\: \sf{T - 2 m_{2}g \: sin \: \theta = m_{2}a ..... (2) }$

☯ Adding both (1) & (2)

$\dashrightarrow\:\: \sf{ m_{1}g - 2 m_{2}g \: sin \: \theta = m_{1}a + m_{2}a }$

$\dashrightarrow\:\: \sf{ ( m_{1} - m_{2} \: sin \: \theta)g =( m_{1} + m_{2})a }$

$\dashrightarrow\:\: \sf{a = \dfrac{( m_{1} - m_{2} \: sin \: \theta)g}{ (m_{1} + m_{2}) } }$

☯ Putting "a" value in (2)

$\dashrightarrow\:\: \sf{T - 2 m_{2}g \: sin \: \theta = m_{2} \times \dfrac{( m_{1} - m_{2}g \: sin \: \theta )g }{ m_{1} + m_{2} } }$

$\dashrightarrow\:\: \sf{T= \frac{( m_{1} - m_{2}g \: sin \: \theta ) m_{2} g}{ m_{1} + m_{2} } + 2 m_{2}g \: sin \: \theta}$