A uniform plank of mass `m = 1kg`, free to move in the horizontal direction only, is placed at the top of a solid cylinder of mass `2m` and radius `R`. The plank is attched to a fixed wall by means of light spring of spring constant `k = 7N//m^(2)`. There is no slipping between the cylinder and the plank, and between the cylinder and the ground. the angular frequency of small oscillations of the system is

Answer

2

Explanation

Suppose that the plank is displaced from its equilibrium position by `x` at time `t`, the centre of the cylinder is, therefore, displaced by `x//2`.
`:.` the mechanical energy of the system is given by,
`K = K.E. ("Plank") +P.E. ("spring") +K.E. ("cylinder")`
`E = (1)/(2)m((dx)/(dt))^(2) +(1)/(2)kx^(2) +(1)/(2)2m {(d)/(dt)((x)/(2))}^(2)`
`+(1)/(2) ((1)/(2)2m.R^(2)) {(1)/(R)(d)/(dt)((x)/(2))}^(2) = (1)/(2) ((7)/(4)m)((dx)/(dt))^(2)+(1)/(2)kx^(2)`
After differentiating the total energy and equating it to zero, one finds acceleration `=- omega^(2)x`
The angular frequency, `omega = sqrt((4k)/(7m)) = 2 rad//sec`