Water flows in a cylindrical, horizontal pipe. As the pipe widens to twice its initial diameter the pressure in the pipe changes.

Calculate the change in pressure between the wide and narrow regions of the pipe. Give your answer as an expression in terms of the density of water, rho, and initial speed v.

The incompressible continuity equation

(Area)*(velocity) = constant
tells you that the speed is 1/4 as high in the wider section. Thus v' = v/4 is the new velocity.

Use the Bernoulli equation to complete the problem. The pressure increase equals the decrease in (1/2)(rho)V^2

Show your work if you need additional assistance.

acertain solid weights 20N in air and 15N when submarged wholly in oil with arclativ density of 0.8 determine a)the volume of oil displaced b)the density of solid

To calculate the change in pressure between the wide and narrow regions of the pipe, we need to use Bernoulli's equation, which relates the pressure, velocity, and height of a fluid flowing in a pipe.

Bernoulli's equation states:
P₁ + 1/2 * ρ * v₁² + ρ * g * h₁ = P₂ + 1/2 * ρ * v₂² + ρ * g * h₂

Where:
P₁ and P₂ are the pressures at two points in the pipe,
ρ is the density of water,
v₁ and v₂ are the velocities at the two points,
g is the acceleration due to gravity,
h₁ and h₂ are the heights of the fluid at the two points.

Assuming the pipe is horizontal, the height difference (h₁ - h₂) can be neglected.

In the narrow region of the pipe, the velocity is v₁, and in the wide region, the velocity is v₂ = v₁/2 (since the diameter doubles, the velocity reduces by a factor of 2 according to the equation of continuity).

Plugging these values into Bernoulli's equation, we get:
P₁ + 1/2 * ρ * v₁² = P₂ + 1/2 * ρ * (v₁/2)²

Expanding and simplifying:
P₁ + 1/2 * ρ * v₁² = P₂ + 1/8 * ρ * v₁²

Bringing all terms to one side:
P₁ - P₂ = 1/8 * ρ * v₁² - 1/2 * ρ * v₁²

Combining like terms:
P₁ - P₂ = 7/8 * ρ * v₁²

Therefore, the change in pressure between the wide and narrow regions of the pipe is given by the expression:
ΔP = P₁ - P₂ = 7/8 * ρ * v₁².

Note that this expression assumes no change in height between the two regions. If there is a significant difference in height, it must be accounted for in the equation.