0625 · 1.4Density

Density formula and units

0:00A small steel block can have more mass than a much larger piece of wood. Size alone does not tell us how much matter an object contains. Density connects an object's mass with the space it occupies. By the end of this video, you will understand density, measure it using three different methods, and use it to predict floating. We need two familiar quantities before we begin. Mass measures the quantity of matter in an object, and we measure it using a balance.

0:24Volume is the amount of space occupied by an object or liquid. Remember that mass is not weight, so a density calculation uses grams or kilograms, not newtons. Density is mass per unit volume. This means the mass contained in each unit of volume, such as each cubic centimetre. Imagine comparing equal-sized pieces of two materials. The material with more mass in that same space has the greater density. Suppose one cubic centimetre of one material has a mass of two grams.

0:53Its density is two grams per cubic centimetre. Another material has eight grams in each cubic centimetre, so its density is eight grams per cubic centimetre. The second material contains four times as much mass in the same volume. To calculate density, we divide the total mass by the total volume. Dividing spreads the mass equally across all the units of volume. For example, twenty grams spread across ten cubic centimetres gives two grams in each cubic centimetre.

1:20That is why density equals mass divided by volume. The unit follows directly from the calculation. Mass in grams divided by volume in cubic centimetres gives grams per cubic centimetre. Mass in kilograms divided by volume in cubic metres gives kilograms per cubic metre. Always check that the units in your calculation match the density unit you need. One gram per cubic centimetre equals one thousand kilograms per cubic metre.

1:44A cubic metre contains one million cubic centimetres, and each one has a mass of one gram. The total mass is therefore one million grams, which is one thousand kilograms. Converting density requires accounting for both the mass unit and the volume unit. Density also lets us calculate a missing mass or volume. Mass equals density multiplied by volume, because each unit of volume contributes its own share of mass.

2:08Volume equals mass divided by density. These relationships describe the same connection, rather than three separate facts to memorise.

Measuring regular solids and liquids

2:16Let us first determine the density of a regularly shaped solid. Measure its mass using a balance. For a rectangular block, measure its length, width and height using a suitable measuring instrument. Its volume equals length multiplied by width multiplied by height, because this counts the space throughout the block. Suppose a block has a mass of one hundred and twenty grams. Its length is five centimetres, its width is three centimetres, and its height is two centimetres.

2:42Multiplying five by three by two gives a volume of thirty cubic centimetres. Dividing one hundred and twenty grams by thirty cubic centimetres gives a density of four grams per cubic centimetre. Notice why we calculated volume before density. Measuring one edge alone does not tell us how much space the whole block occupies. All three dimensions are needed for a rectangular block. For another regular shape, use the appropriate geometrical formula for its volume, then divide its mass by that volume.

3:10For a liquid, we need its mass without including the container. Measure the mass of an empty, dry measuring cylinder. Add the liquid, then measure the mass of the cylinder and liquid together. Subtract the empty cylinder's mass to find the mass of the liquid alone. Read the liquid's volume from the measuring cylinder at eye level. For water and similar liquids, read the bottom of the meniscus. The meniscus is the curved liquid surface.

3:33Reading from above or below can make the level appear incorrectly aligned with the scale. Suppose the empty cylinder has a mass of forty grams. The cylinder and liquid together have a mass of one hundred and four grams. Subtracting forty from one hundred and four gives a liquid mass of sixty-four grams. The measured liquid volume is eighty cubic centimetres, so 64 ÷ 80 gives zero point eight grams per cubic centimetre.

3:56Including the cylinder's mass would give a density that is too high. You would be dividing the mass of two things by the volume of only the liquid. Alternatively, zero the balance with the empty cylinder already on it. The balance then displays the added liquid's mass directly.

Irregular solids and floating

4:11An irregular solid does not have convenient dimensions for a simple volume formula. Instead, we can determine its volume by displacement. When a solid is fully submerged, it occupies space that the liquid previously occupied. The volume of liquid displaced equals the submerged solid's volume. Measure the dry solid's mass using a balance first. Record the initial liquid volume in a measuring cylinder, then carefully lower the solid into the liquid.

4:36Make sure it sinks, is completely submerged, and has no trapped air bubbles. Subtract the initial reading from the final reading to find the solid's volume. Suppose a stone has a mass of fifty-four grams. The water reading starts at fifty cubic centimetres and rises to sixty-eight cubic centimetres. Subtracting fifty from sixty-eight gives a stone volume of eighteen cubic centimetres. Dividing fifty-four grams by eighteen cubic centimetres gives a density of three grams per cubic centimetre.

5:03The final reading includes both the original water and the space occupied by the stone. That is why using sixty-eight cubic centimetres as the stone's volume would be wrong. Air bubbles also occupy space and would make the measured volume too large. Use a liquid that does not dissolve or react with the solid. Now we can use density to predict whether an object floats. An object less dense than the liquid floats, while an object more dense than the liquid sinks.

5:30Compare their densities using the same units. For an object containing empty spaces, use its overall density rather than only the density of its solid material. The underlying idea is that the liquid provides an upward supporting force. For equal volumes, a less dense object has less mass than the liquid it displaces when fully submerged. It can therefore float with only part of its volume submerged. A denser object cannot receive enough support even when fully submerged, so it sinks.

5:57For these comparisons, water has a density of approximately one gram per cubic centimetre. A solid block with a density of zero point six grams per cubic centimetre floats in water. The stone we measured has a density of three grams per cubic centimetre, so it sinks. An object with exactly the liquid's density can remain suspended when fully submerged. A greater mass does not automatically mean an object will sink.

6:19Increasing the size of a uniform material increases both its mass and its volume. Their ratio stays the same, so its density remains unchanged. Floating depends on density compared with the liquid, not simply on how large or massive the object is.

Liquid layers and exam answers

6:33This next idea is Supplement content for Extended students. When two liquids do not mix, the less dense liquid floats on the denser liquid. They form separate layers, with the greater density underneath. The condition that they do not mix matters, because mixed liquids do not remain as distinct layers. Here is an exam-style question. Two liquids that do not mix have densities of zero point eight and one point two grams per cubic centimetre.

6:57You must say which liquid floats on top, and give a reason. The liquid with density zero point eight grams per cubic centimetre floats on the other liquid. It has the lower density, and the liquids do not mix. An examination may also ask you to describe how to determine an irregular solid's density. Measure its mass using a balance. Determine its volume from the rise in liquid volume when it is completely submerged.

7:19Calculate density by dividing its mass by its displaced volume, and state the appropriate unit. Let us bring the ideas together. Density is mass per unit volume, so it compares how much mass occupies the same amount of space. Measure mass with a balance, and choose a volume method suited to the object or liquid. Divide mass by volume, then compare densities to predict floating or the order of separate liquid layers.