A system consisting of a smooth movable wedge of angle `alpha` and a block A of mass m are connected together with a massless spring of spring constant k, as shown in the figure. The system is kept on a frictionless horizontal plane. If the block is displaced slighlly from equilibrium and left to oscillate, find the frequency of small oscillations.

Explanation

Suppose that the relative displacement of the block A w.r.t the wedge is x at same time t, and the corresponding displacement of the block along the horizontal is x.. The total energy of the system can be written as

`E = (1)/(2) M ((dx.)/(dt))^(2) + (1)/(2) m [((dx)/(dt)) cos alpha - ((dx)/(dt))^(2) sin^(2) alpha]` The equation of conservation of momentum gives,
`M ((dx.)/(dt)) = m ((dx)/(dt) cos alpha - (dx.)/(dt))`...(ii)
which gives
`(dx.)/(dt) = (m)/(M +m) ((dx)/(dt)) cos alpha`
and substituting in the equation for total energy gives: Where a represents the extension of the spring when it is in the mean position. The angular frequency `omega` is given by
`omega = sqrt((k)/(m_("red") cos^(2) alpha + m sin^(2) alpha)) " where " m_("red") = (mM)/(m + M)`