A system consists of three masses `m_1, m_2` and my connected by a string passing over a pulley P. The mass `m_1` hangs freely and `m_2` and `m_3` are on a rough horizontal table (the coefficient of friction = `mu`). The pulley is frictionless and of negligible mass. The downward acceleration of mass m, is mi (Assume `m_1 = m_2 = m_3 = m`)

Options

A. `(g (1- gmu))/(9)`
B. `(2 gmu)/(3)`
C. `(g (1- 2 mu ))/( 3)`
D. `(g (1- 2 mu ))/(2)`

Answer

C

Explanation

Force of friction on mass
` m_2 = mu m_2 g `
force of friction on mass
` m_3 = mu m_3 g `
force of friction on mass
`m_3 = mu m_3 g `
let a be common acceleration of the system
` therefore a = (m_1 g - mu m_2 g - mu m_3 g )/( m_1 +m_2 +m_3`
here ` m_1 = m_2 -m_3 -m`
` therefore a= (mg - mu g - mu g )/( m+m+m) = ( mg - 2 mu mg )/( 3m) = (g (1-2 mu))/(3)`
hence the downward acceleration of mass `m_1 ` is ` ( g (1-2 mu))/(3)`