Block B slides on frictionless inclined plane of Block A. Block A also slides on frictionless ground. A rope at one end is attahed to the block B and passes over a pulley attched to block A. The other end of the rope is attached to an immovable wall. Find the relative acceleration of B w.r.t. A and acceleration of ‘A’

Dear Student, please see the diagram below.

Let,

x₁ = distance of block A from wall

x₂ = distance of block B from pulley

L = length of the string.

a = magnitude of acceleration of the block A with respect to ground

b = magnitude of acceleration of the block B with respect to block A

θ = angle of the inclination of the plane.

Then x₁ + x₂ = L

Differentiating twice with respect to time,

we get

Now from FBD of block B (from the reference frame of block A),

we have following forces acting on B,

N = normal reaction

m_Bg = gravitational force

m_Ba = pseudo force

as shown in the figure.

Writing Newton's second law along the plane, we get

m_Bb = m_Bacosθ + m_Bgsinθ

b = acosθ + gsinθ