Askiitians Tutor Team
Last Activity: 5 Months ago
To solve the problem of how the flow rate of water changes when the radius of a capillary tube is altered, we can apply principles from fluid dynamics, specifically Poiseuille's law. This law describes how the flow rate of a fluid through a cylindrical pipe is influenced by several factors, including the radius of the pipe, the length of the pipe, the viscosity of the fluid, and the pressure difference driving the flow.
Understanding Poiseuille's Law
Poiseuille's law states that the volumetric flow rate (Q) of a fluid through a cylindrical pipe is given by the formula:
Q = (π * r^4 * ΔP) / (8 * η * L)
Where:
- Q = volumetric flow rate (in cubic centimeters per second, cm³/s)
- r = radius of the tube (in centimeters)
- ΔP = pressure difference (in dynes/cm²)
- η = dynamic viscosity of the fluid (in poise)
- L = length of the tube (in centimeters)
Initial Conditions
In the first scenario, we have a capillary tube with radius a and length L, through which 16 cm³ of water flows per second. This means:
Q₁ = 16 cm³/s
Changing the Radius
Now, we consider a second tube with the same length L but with a radius of a/2. According to Poiseuille's law, the flow rate will depend on the fourth power of the radius:
Q₂ = (π * (a/2)^4 * ΔP) / (8 * η * L)
Calculating the New Flow Rate
We can express the new flow rate in terms of the original flow rate:
Q₂ = (π * (a/2)^4 * ΔP) / (8 * η * L) = (π * (a^4 / 16) * ΔP) / (8 * η * L)
Now, we can relate this back to the original flow rate:
Q₂ = (1/16) * (π * a^4 * ΔP) / (8 * η * L) = (1/16) * Q₁
Since we know that Q₁ = 16 cm³/s, we can substitute this value into our equation:
Q₂ = (1/16) * 16 cm³/s = 1 cm³/s
Final Result
Therefore, when the radius of the tube is halved, the quantity of water flowing through the tube per second will be 1 cm³/s.
