you need to extend a 2.50 inch diameter pipe, but you only have 1.00 inch diameter pipe on hand.you make a fitting to connect these pipes end to end. if the water is flowing at 6.00cm/s in the wide pipe, how fast will it be flowing through the narrow one?
Oh, pipe problems, huh? Well, it seems like you've got quite the situation on your hands! Trying to fit a 2.50-inch pipe with a 1.00-inch pipe is like trying to squeeze a sumo wrestler into a pair of skinny jeans – it's just not gonna work!
But let's dive into some math humor to help solve this conundrum. If we want to find out how fast the water will flow through the narrow 1.00-inch pipe, we can use a little thing called the "Principle of Cross-Sectional Area Conservation." Fancy, huh?
Since the area of a circle is proportional to the square of its diameter, we can say that the ratio of the areas of the wide pipe to the narrow pipe is equal to the square of the ratio of their diameters. So, (2.50/1.00)^2 should give us the answer!
Let me crunch the numbers for you... *calculating sounds*
Okay, here it is... *drumroll* The water will flow about 6.25 times faster through the narrow 1.00-inch pipe than it does through the wide 2.50-inch pipe.
So, get ready for some fast-flowing water! Just make sure you don't sprout any water-powered rocket boots while you're at it. Safety first, my friend!
To find the flow rate of water through the narrow 1.00 inch diameter pipe, we can use the principle of continuity. According to the principle of continuity, the flow rate of an incompressible fluid remains constant through a pipe.
Here are the steps to calculate the flow rate:
Step 1: Calculate the cross-sectional area of each pipe.
The cross-sectional area (A) of a pipe is given by the formula A = πr^2, where r is the radius of the pipe.
For the wide pipe with a 2.50 inch diameter:
Radius (r) = Diameter / 2 = 2.50 inches / 2 = 1.25 inches
Convert the radius to centimeters:
1 inch = 2.54 cm
Radius = 1.25 inches * 2.54 cm/inch = 3.175 cm
The cross-sectional area of the wide pipe is then:
A_wide = π(3.175 cm)^2
For the narrow pipe with a 1.00 inch diameter:
Radius = 1.00 inches / 2 = 0.50 inches
Convert the radius to centimeters:
Radius = 0.50 inches * 2.54 cm/inch = 1.27 cm
The cross-sectional area of the narrow pipe is:
A_narrow = π(1.27 cm)^2
Step 2: Calculate the flow rate in the wide pipe.
Given the water flow velocity in the wide pipe, v_wide = 6.00 cm/s
The flow rate in the wide pipe can be calculated using the formula:
Q_wide = A_wide * v_wide
Step 3: Calculate the flow rate in the narrow pipe.
The flow rate in the narrow pipe is the same as in the wide pipe (due to the principle of continuity). Therefore:
Q_narrow = Q_wide = A_wide * v_wide
Step 4: Find the flow velocity in the narrow pipe.
We can rearrange the flow rate formula to find the flow velocity in the narrow pipe:
v_narrow = Q_narrow / A_narrow
Substituting the values:
v_narrow = (A_wide * v_wide) / A_narrow
Finally, plug in the values for A_wide and A_narrow, and calculate the flow velocity in the narrow pipe.
Please note that the answer will depend on the exact values calculated using the equations above.
To determine how fast the water will be flowing through the narrow pipe, you can use the principle of continuity, which states that the volume flow rate of an incompressible fluid remains constant within a closed system.
The volume flow rate can be calculated by multiplying the cross-sectional area of the pipe by the velocity of the fluid. Since the pipes are connected end to end, their combined volume flow rates must be equal.
Here's how you can calculate the velocity of the water flowing through the narrow pipe:
1. Determine the cross-sectional area of the wide pipe:
- The diameter of the wide pipe is given as 2.50 inches. Radius (r) can be calculated by dividing the diameter by 2: r = 2.50 inches / 2 = 1.25 inches.
- Convert the radius to centimeters: 1.25 inches * 2.54 cm/inch = 3.175 cm.
- Calculate the cross-sectional area using the formula: A = π * r^2 = 3.14 * (3.175 cm)^2 ≈ 31.79 cm^2.
2. Determine the velocity of the water in the wide pipe:
- The given velocity is 6.00 cm/s.
3. Determine the cross-sectional area of the narrow pipe:
- The diameter of the narrow pipe is given as 1.00 inch. Using the same process as above, we find that the radius is approximately 1.27 cm.
- Calculate the cross-sectional area using the formula: A = π * r^2 = 3.14 * (1.27 cm)^2 ≈ 5.06 cm^2.
4. Calculate the velocity of the water in the narrow pipe:
- Since the volume flow rate is constant, we can set up an equation: A_wide * v_wide = A_narrow * v_narrow.
- Plug in the known values: (31.79 cm^2) * (6.00 cm/s) = (5.06 cm^2) * (v_narrow).
- Solve for v_narrow: v_narrow ≈ (31.79 cm^2 * 6.00 cm/s) / 5.06 cm^2 ≈ 37.8 cm/s.
Therefore, the water will be flowing through the narrow pipe at approximately 37.8 cm/s.
According to what is called the continuity equation, the product of
(density)(cross sectional area)(velocity)
is a constant in steady-state flow.
For a liquid such as water, you can assume that the density is constant.
Therefore A*V = constant
Since Area is proprtional to diameter ^2,
D^2*V = constant
Now do the numbers.