Find out tension in the rop:

Two blocks A and B are associated as shown in Figure (a). Mass of block A is 15 kg and this is resting on a surface having a coefficient of friction μ = 0.30. The mass of block B is equal to7 kg. The pulley is frictionless. Find out the acceleration of the mass B and tension in the rope.

Solution

As the pulley is frictionless, the tension in the string shall be same and acceleration of A and B also shall be same. Let a be the acceleration and T the tension in the string.

Consider Free Body Diagram of both the bodies as shown in Figure. Consider equilibrium of body A.

∑ F_x  = 0, T = 15 a + μ N₁

                    = 15 a + 0.3 N₁ ----------- (1)

∑ F_y  = 0, N₁ = 15 × 9.81

                     = 147.15 N            ----------- (2)

By Substituting in Eq. (1), we obtain

T = 15 a + 0.3 × 147.15

= 15 a + 44.145           ----------- (3)

 Now let us assume equilibrium of body B.

∑ F_y  = 0,                    T + 7a = 7 × 9.81

              T = 68.67 - 7 a            ------------- (4)

From Eqs. (3) and (4), we obtain

T = 15 a + 44.145 = 68.67 - 7 a

∴ 22 a = 24.525

∴ a = 1.115 m / sec.

By Substituting this value of a in either Eqs. (3) or (4), we obtain

T = 15 a + 44.145

= 15 × 1.115 + 44.145 = 60.87 N