Density of a Stone

Date: 09/14/2002 at 10:54:02
From: Amanda
Subject: Density

Hello, Dr. Math.

Can you please explain this question to me?

A stone having a mass of 160 g displaces the level of water in a 
cylinder by 30 mL. Find the density of the stone.

Thank you.

Date: 09/14/2002 at 11:20:44
From: Doctor Ian
Subject: Re: Density

Hi Amanda,

'Density' is a little like speed, in that it relates two quantities. 

Suppose there are two cars, and one goes 2 miles in one minute, while 
the other goes only 1 mile. The speed of the first is 2 miles per 
minute, and the speed of the second is 1 mile per minute. The compound 
units 'miles per minute' are used to indicate a particular speed. 

Now think about two stones with exactly the same volume. (One way to 
measure the volume of something is to dunk it in water, and see how 
much of the water has to move out of the way to make room for it.)  
Let's say each stone displaces two liters of water. The first stone 
has a mass of 10 kg, and the other has a mass of 20 kg. 

The second one packs more mass into the same space. When that happens, 
we say that the second one is 'more dense', or has 'greater density' 
than the first, in the same way that we say that one of the cars is 
faster than the other. The units of density are 'mass per volume'. 

So we'd say the first stone has a density of 10 kg per 2 liters, while 
the second has a density of 20 kg per 2 liters. We would normally 
simplify that, 

  20 kg / 2 liters = (20/2) kg/liter

                   = 10 kg/liter

in the same way that we might simplify the calculation of a speed, 

  30 miles / 15 minutes = (30/15) miles/minute

                        = 2 miles/minute  

Note that in both cases, you can use various units for the individual 
quantities. Speed might be measured in miles per minute, kilometers 
per second, centimeters per week, and so on. Similarly, density might 
be measured in grams per liter, kilograms per milliliter, ounces per 
gallon, and so on. 

There is another way that density is like speed. When you know a 
distance and a time, you can compute a speed:

  10 miles / 2 hours = (10/2) miles/hour = 5 miles/hour

But from a speed and a distance, you can compute a time:

  10 miles / (5 miles/hour) = (10/5) hours = 2 hours

And from a speed and a time, you can compute a distance:

  2 hours * 5 miles/hour = (2*5) miles = 10 miles

Similarly, if you have a mass and a volume, you can compute a density:
 
 
  density = mass / volume
 
 
But if you have a density and a mass, you can compute a volume:
 
 
  mass / density = volume
 
 
And if you have a density and a volume, you can compute a mass:
 
 
  density * volume = mass 

It's really just the same kind of idea applied to two different kinds 
of measurements.  

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/