All the surfaces shown in the figure are assumed to be frictionless. The block of mass m slides on the prism which in turn slides backwards on the horizontal surface. Find the acceleration of the smaller block with respect to the prism.
Assuming the wedge moves with a
Assuming the block moves with a’
acceleration component for gravity = g sinø
Wedges acceleration component = a cosø
a cosø + g sinø = a’ ( relative to each other ) -(1)
For other component normal-
acceleration component for gravity (y axis)= g cosø
Wedges acceleration component (y axis) = a sinø
(a sinø – g cosø)m = N – (2)
Looking at the wedge the Normal has two components
in the x axis the normal is Nsinø = aM
replacing N in equation (2)
aM/sinø = m(a sinø – g cosø)
a (M/sinø – msinø ) = mg cosø
a ( M – m sin^2ø/sinø) = mg cosø
a (M – m sin^2ø) = mg cosøsinø
a = mg cosøsinø/ M – m sin^2ø