Question

An incompressible fluid flows steadily through a cylindrical pipe which has radius 2

R 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_____
A. 2v
B. v
C. v/2
D. 4v

Answer 1

The velocity of an incompressible fluid at point B is 4 times the velocity at point A due to the conservation of mass, which makes the correct answer (D) 4v.

Step-by-step explanation:

The question concerns the principle of conservation of mass as it applies to the steady flow of an incompressible fluid through a cylindrical pipe with varying radii. According to this principle, for an incompressible fluid, the rate of fluid flow (or flow rate) must be constant throughout the pipe. The flow rate Q is given by the equation Q = AØu, where A is the cross-sectional area, and u is the average velocity of the fluid.

At point A, the radius of the pipe is 2R, so the cross-sectional area is A1 = π(2R)^2 = 4πR^2. At point B, the radius is R, giving a cross-sectional area A2 = πR^2. Applying the continuity equation A1u1 = A2u2, and substituting in the known values and the velocity at point A, v, we get 4πR^2v = πR^2u2, which simplifies to u2 = 4v. Therefore, the velocity at point B is four times the velocity at point A, making the correct answer (D) 4v.