Question

An incompressible fluid flows steadily through a cylindrical pipe which has radius 2R at point A and radius R at point B farther along the flow direction. If the velocity at point A is v, its velocity at point B is

Answer 1

The velocity at point B is 4v, due to the conservation of mass for incompressible fluids, which requires that the flow rate is constant throughout a pipe, leading to an increase in velocity when the cross-sectional area decreases.

Step-by-step explanation:

The question deals with the principle of conservation of mass for an incompressible fluid flowing through a pipe with varying cross-sectional areas. According to the continuity equation, the product of cross-sectional area (A) and velocity (v) at any two points in the flow must be constant (Q = Av). For a cylindrical pipe, the cross-sectional area is π times the radius squared (πr2).

At point A, the radius is 2R and the velocity is v, so the flow rate Q is π(2R)2v. At point B, the radius is R, and the flow rate must remain the same; hence the velocity at point B must increase to 4v to compensate for the smaller area. This is consistent with the principle that the flow rate (product of area and velocity) must be the same at all points along the pipe for an incompressible fluid.