Question

How do you find the change in volume produced by a piston if area of the piston= 1*10^(-16)m^2 and the distance moved by the piston is 85 nm? ∆V = A*∆d

Answer 1

Hi! It looks like you have a good equation! It's just a matter of making the units agree and plugging in values! Would you agree?

Answer 2

Yes, but when I multiply the two I get some weird answer like 8.5E-15. I asked my teacher about this and he told me to pay attention to my units, that the area is measured in cubic units. What would I do then?

Answer 3

Well, the area is measure in square units of distance, and the volume is measured in cubic units of distance.

Like,

Answer 4

So that's that.

But you have an issue where your units are \(m^2\) and \(nm\). So you've multiplied them and now have units of \(m^2\ nm\)

Answer 5

While \(8.5\times 10^{-15}\ [m^2\ nm]\) is correct, it's not a very useful amount.

When you have different units, you can change the units before or after you do the calculation. I'll show you how. Which would you prefer - before, or after?

Answer 6

I'll go with before.

Answer 7

Alright! Now, \(1\times 10^{-16} [m^2]\) is very small. We could convert it to \(nm^2\). Or, we could change the \(85\ nm\) to be in meters. Up to you!

Answer 8

Ohh I see, so how would you change 85 nm to meters?

Answer 9

You can see that there's a lot of flexibility with these conversions. However, the basic step is the same.

Say you wanted to change \(85\ nm\) to meters. You might know that \(10^{-9}\ [m]=1\ [nm]\).
I'll show you the reasoning, too.
You can use algebra to see that \[10^{-9}\ [m]=1\ [nm]\\\qquad\Downarrow\\\frac{10^{-9}\ [m]}{1\ [nm]}=1\]
Since it's just 1, you can multiply it by anything without changing the actual value.
So you multiply it by the unit you want to convert. That's the \(85\ nm\).
\[85\ [nm] \times \frac{10^{-9}\ [m]}{1\ [nm]} = 85\ [\cancel{nm}] \times \frac{10^{-9}\ [m]}{1\ [\cancel{nm}]} = 85\times 10^{-9}\ [m]\]

Answer 10

So the change in voulme would be 1*10^(-16)m^2 * 85*10^-9m?

Answer 11

Yep!

Answer 12

Congrats!

And that was easier than changing the squared units. But here's a trick to discovering that conversion in case you can't find it, or it's a test.\[m^2=m\times m\]and\[10^{-9}\ m=1\ nm\]so, by squaring both sides,\[10^{-9}\ m\times 10^{-9}\ m = 1\ nm\times 1\ nm\]which is\[10^{-18}\ m^2 = 1\ nm^2\] after calculation.

Answer 13

Similarly, you could discover that \(10^{-27}\ m^3=1\ nm^3\), and so your answer of \(8.5\times 10^{-24}\ [m^3]\) is.....\[8.5\times 10^{-24}\ [m^3]\times \frac{1\ [nm^3]}{10^{-27}\ [m^3]}=8.5\times 10^{3}\ [nm^3]\]

Answer 14

Alright thanks so much for your help!

Answer 15

You're welcome! Good luck with anything you have to do!

Answer 16

would the final answer be 85m^3*10^-25??

Answer 17

Yep!

Answer 18

If you saw a discrepancy, it was probably that mine included a decimal, and the power had to change to keep the same value.