According to the principle of continuity, the velocity at the exit of the pipe when the diameter reduces to 0.5d is 4 times the initial velocity.
To find the velocity at the exit of the pipe when the diameter reduces to 0.5d, we can use the principle of continuity. According to the principle of continuity, the volume flow rate of an incompressible fluid remains constant along a pipe. The volume flow rate (Q) is given by Q = Av, where A is the cross-sectional area of the pipe and v is the velocity of the fluid.
If the diameter of the pipe reduces to 0.5d, the cross-sectional area of the pipe is reduced by a factor of (0.5d/d)² = 0.25.
Since the volume flow rate is constant, we can write (A₁)(v₁) = (A₂)(v₂), where A₁ is the initial cross-sectional area, v₁ is the initial velocity, A₂ is the final cross-sectional area, and v₂ is the velocity at the exit of the pipe. Substituting the values, we get:
(A₁)(v₁) = (A₂)(v₂) (1)
Since A₂ = (0.25)(A₁), we can substitute this value in equation (1) and solve for v₂:
(A₁)(v₁) = (0.25)(A₁)(v₂)
v₂ = (1/0.25)(v₁)
v₂ = 4(v₁)
Therefore, the velocity at the exit of the pipe is 4 times the initial velocity.