Find out the motion of the system:

Block (A) of mass M _A = 40 kg is resting on coarse table having coefficient of friction = 0.1. This is tied up with an inextensible rope (a b c d) passing over the smooth and mass-less pulleys P₁ and P₂ and end (d) is at the fixed support is illustrated in Figure. Another mass M_B  = 10 kg is hanging from pulley P₂.Find out the motion of the system.

Solution

Figure illustrated the block A and B as connected in the system. Pulley P₂ has rope on its either side or therefore movement of P₂ or block B is half the movement of block A.

∴ If V is the velocity of block A, V/2 is the velocity of block B.

Likewise, if a is the acceleration of A, a/2 is the acceleration of B.

∴          For block A, V ² = 2 a s                    --------- (I)

Weight of B = W_B  = 10 × g

Let A′ and B′ be the final positions of block A and B respectively as illustrated in the figure.

Let Datum-level at the lowest level B′ of the block B.

40 g ( H + (s/2) )  + 10 (s/2) × g = 40 g (H + (s/2) + M _A (V ² /2) +   (M _B/2) (V /2)² + F × s

∴ 5 g s = (V ²/2) (40 +( 10/4) )+ 0.1 × 40 g s                  ---------- (II)

From Eq. (I) and (II)

5 g s - 4 g s = (2 a/2) s (42.5)

∴ a = 0.23 m / sec ² along AA.