QuestionIn a pipeline with a 30 cm diameter and water speed of 1.5 m/s, find the speed at a 1 cm diameter.

Studdy Solution

STEP 1

Assumptions1. The diameter at the first point in the pipeline is30 cm. . The speed of water flow at the first point is1.5 m/s.
3. The diameter at the second point in the pipeline is1 cm.
4. The speed of water flow at the second point is what we need to find.
5. The flow of water in the pipeline is steady, meaning the volume of water entering a section of the pipeline per unit time is equal to the volume of water leaving that section per unit time. This is also known as the principle of continuity in fluid dynamics.

STEP 2

First, we need to convert the diameters from cm to m, since the speed of water flow is given in m/s. $iameter_{1} =30 cm =0. m$ $iameter_{2} =1 cm =0.01 m$

STEP 3

We use the principle of continuity, which states that the product of the cross-sectional area and the speed of flow at one point in the pipeline is equal to the product of the cross-sectional area and the speed of flow at another point. Mathematically, this is expressed as $Area_{1} \times Speed_{1} = Area_{2} \times Speed_{2}$

STEP 4

The cross-sectional area of a pipe is given by the formula for the area of a circle, which is $\pi r^{2}$ , where $r$ is the radius of the pipe. Since the diameter is twice the radius, we can express the area in terms of the diameter as follows $Area = \pi (iameter/2)^{2}$

STEP 5

Substitute the formula for the area into the equation from3 $\pi (iameter_{1}/2)^{2} \times Speed_{1} = \pi (iameter_{2}/2)^{2} \times Speed_{2}$

STEP 6

We can cancel out the $\pi$ on both sides of the equation, and we are left with $(iameter_{1}/2)^{2} \times Speed_{1} = (iameter_{2}/2)^{2} \times Speed_{2}$

STEP 7

Rearrange the equation to solve for $Speed_{2}$ $Speed_{2} = \frac{(iameter_{1}/2)^{2} \times Speed_{1}}{(iameter_{2}/2)^{2}}$

STEP 8

Substitute the given values for $iameter_{1}$ , $Speed_{1}$ , and $iameter_{2}$ into the equation $Speed_{2} = \frac{(0.3/2)^{2} \times1.5}{(0.01/2)^{2}}$

STEP 9

Calculate the value of $Speed_{2}$ $Speed_{2} = \frac{(.15)^{2} \times.5}{(.005)^{2}} =135 m/s$ The speed of flow at a point where the diameter is cm is135 m/s. The closest answer choice to this is D.) $135 \times^{\wedge}3 \mathrm{~m} / \mathrm{s}$ , so this is the correct answer.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms