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Chapter 9: Problem 70

Assume a water strider has a roughly circular foot of radius $0.02 \mathrm{mm} .$ (a) What is the maximum possible upward force on the foot due to surface tension of the water? (b) What is the maximum mass of this water strider so that it can keep from breaking through the water surface? The strider has six legs.

Short Answer

Expert verified
Answer: The maximum upward force on each foot due to surface tension of the water is \(0.00288 \pi\) N, and the maximum mass of the water strider so that it doesn't break through the water surface is approximately 5.56 g.

Step by step solution

01

Surface Tension Force Formula

Recall the formula for the force exerted by surface tension is given by: $$F = 2\pi r T$$ where \(F\) is the force exerted by surface tension, \(r\) is the radius of the circular foot, and \(T\) is the surface tension of the water. The surface tension of water at room temperature is approximately \(T = 72 \times 10^{-3} \mathrm{N/m}\).
02

Calculate the Upward Force on Each Foot

Substituting the given radius and surface tension of water into the formula, we can find the maximum upward force on each foot: $$F = 2\pi (0.02 \times 10^{-3} \mathrm{m}) (72 \times 10^{-3} \mathrm{N/m})$$
03

Simplify the Expression

Now, solve for \(F\) and simplify the expression: $$F = 2\pi (0.02 \times 10^{-3}) (72 \times 10^{-3}) = 0.00288 \pi \mathrm{N}$$
04

Calculate the Upward Force on All Six Feet

Since the water strider has six legs, multiply the force on one foot by six to find the total upward force on all six feet: $$F_{total} = 6 \times 0.00288 \pi \mathrm{N} = 0.01728 \pi \mathrm{N}$$
05

Maximum Mass of the Water Strider

To find the maximum mass of the water strider, we need to consider that this total upward force must be equal to the gravitational force acting on the water strider to prevent it from breaking through the water surface. The gravitational force on the strider can be calculated as: $$F_g = mg$$ where \(m\) is the mass of the water strider and \(g\) is the acceleration due to gravity (approximately \(g = 9.81 \mathrm{m/s^2}\)). Set the total upward force equal to the gravitational force and solve for \(m\): $$0.01728 \pi \mathrm{N} = mg$$ $$m = \frac{0.01728 \pi \mathrm{N}}{9.81 \mathrm{m/s^2}}$$
06

Final Answer

Finally, calculate the maximum mass \(m\) and convert it to grams (since 1 kg = 1000 g) for a more suitable unit: $$m = \frac{0.01728 \pi}{9.81} \mathrm{kg} \approx 5.56 \times 10^{-3} \mathrm{kg} = 5.56 \mathrm{g}$$ So, the answers are: (a) The maximum possible upward force on each foot due to surface tension of the water is \(0.00288 \pi\) N. (b) The maximum mass of the water strider so that it can keep from breaking through the water surface is approximately 5.56 g.

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