Physics calc/surface area

**BiGpO6790** · August 30th 2009, 08:52 PM

Grains of fine California beach sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide, which has a density of 2.8 × 103 kg/m3. What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube 1.1 m on an edge?

I have calculated the surface areas of the sand but i am stumped from there on... please help.

**pedrosorio** · August 31st 2009, 01:13 AM

$\mbox{Surface Area of N grains of sand}= 4 \pi r^2 N$ (1)

$\mbox{Surface Area of a 1.1m edge cube} =6 \cdot 1.1^2$ (2)

$\mbox{Mass of N grains of sand} = \mbox{Density of sand} \cdot \mbox{Volume of N grains of sand}$ (3)

$\mbox{Volume of N grains of sand} = \frac{4}{3} \pi r^3 N$ (4)

So, (1)=(2):

$4 \pi r^2 N = 6 \cdot 1.1^2$

$N = \frac{3 \cdot 1.1^2}{2\pi r^2}$ (5)

Now we compute the volume using (4) and replacing N with (5):

$Volume = \frac{4 \pi r^3 \cdot 3 \cdot 1.1^2}{3 \cdot 2 \pi r^2}$

$Volume = 2 r \cdot 1.1^2$ (6)

Finally we compute the mass using (3) and (6):

$Mass = 2.8 \cdot 10^3 \cdot 2 r \cdot 1.1^2$

Be aware that I assume r is in meters (therefore you'll have to convert 50 micrometers to meters), the final mass comes in kg.

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