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Chapter 2: Problem 16

Often young children drink milk \(\left(\rho=1030 \mathrm{kg} / \mathrm{m}^{3}\right)\) through a straw. Determine the maximun length of a vertical straw that a child can use to empty a milk container, assuming that the child can develop \(75 \mathrm{mm}\) Hg of suction, and use this answer to determine if you think this is a reasonable estimate of the suction that a child can develop.

Short Answer

Expert verified
The maximum length of a vertical straw a child can use to empty a milk container, assuming that the child can develop \(75 \) mm Hg of suction power, is approximately \( 99 \) cm which is a reasonable estimate of the suction that a child can develop.

Step by step solution

01

Understand the Problem

The child creates a vacuum in their mouth, lowering the pressure at the top of the straw. The outside air pressure is then higher than pressure in the straw, and pushes the milk up the straw. To know the highest milk can be sucked up the straw, it's necessary to calculate the maximum height the milk can be pushed by atmospheric pressure.
02

Convert Units

Given suction power of child is \(75 \) mm Hg. Convert this suction power to Pascal - \(75 \) mm Hg = \(75 \times 133.32 \) Pa. Calculating this, it gives us a suction power of \(10,000 \) Pa. Here we have used the equivalence \(1 \) mm Hg = \(133.32 \) Pa.
03

Using Fluid Pressure Formula

Pressure at a depth in a fluid column is given by \( P = \rho g h \), where \( \rho \) = fluid density, \( g \) = acceleration due to gravity and \( h \) = height of the fluid column. Solving for \( h \), we get \( h = P/(\rho g) \).
04

Calculation

Inserting the values into the formula, for \( P = 10,000 \) Pa, \( \rho = 1030 \) kg/m³ (milk's density), and \( g = 9.8 \) m/s² (approx value of gravity), the \( h = 10,000 / (1030 \times 9.8) \). Calculating this, we get \( h \approx 0.99 \) m. This is the maximum height the milk can be pushed up.
05

Validating the Answer

Since 0.99 m (or 99 cm) is relatively taller than the usual length of a milk straw which is about 20-25 cm, this largely exhibits a child's suction ability. Hence, this seems a reasonable estimate of the suction that a child can develop.

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