Video Transcript
In this video, we will learn how to
calculate the density of an object given its mass and volume and also convert
between different units of density. We will begin by explaining what we
mean by density and recall the formula we can use to calculate it.
Density is a way of comparing how
heavy a material is for its size. It is a measurement of the amount
of a substance contained in a certain volume. The density of any solid is the
mass of the object divided by its volume. This leads us to the general
formula density is equal to mass divided by volume.
The mass of an object is usually
measured in kilograms or grams. The volume is measured in cubic
meters or cubic centimeters. This leads us to two standard units
for density, either kilograms per cubic meter or grams per cubic centimeter. The general formula density is
equal to mass divided by volume is sometimes represented in the triangle shown. This means that we can also
calculate the mass by multiplying the density by the volume or calculate the volume
by dividing the mass by the density.
We will now use this formula to
calculate the density of an object given its mass and volume.
True or false: A cylinder with a
volume of one sixtieth of a cubic meter and a mass of 150 kilograms has a density of
9,000 kilograms per cubic meter.
We recall that the density of any
object is equal to its mass divided by its volume. If the mass is measured in
kilograms and the volume in cubic meters, then the units for density will be
kilograms per cubic meter. In this question, we are told the
cylinder has a mass of 150 kilograms and a volume of one sixtieth of a cubic
meter. The density will therefore be equal
to 150 divided by one sixtieth. Dividing by a fraction is the same
as multiplying by the reciprocal of the fraction. This is sometimes known as KCF. We keep the first number the same,
we change the sign to a multiplication, and we flip the fraction.
60 divided by one is equal to
60. So the density is equal to 150
multiplied by 60. 15 multiplied by six is equal to
90. This means that 150 multiplied by
60 is equal to 9,000. The density of the cylinder is
9,000 kilograms per cubic meter. As this is the value we were given
in the statement, the correct answer is true. A cylinder with a volume of one
sixtieth of a cubic meter and a mass of 150 kilograms will have a density of 9,000
kilograms per cubic meter.
In our next question, we will
calculate the density of a cube.
A cube has a side length of 0.15
meters. If the mass of the cube is three
kilograms, what is its density? Give your answer approximated to
one decimal place if required.
Let’s begin by considering a cube
which has side length 0.15 meters. The volume of any cube can be
calculated by cubing the side length. In this question, we need to cube
0.15. This is equal to 0.003375. We also know that if the side
length is in meters, the volume will be in cubic meters.
We have been asked to calculate the
density of the cube. And we know that density is equal
to mass divided by volume. As the mass is equal to three
kilograms, the density will be equal to three divided by 0.003375. Typing this into the calculator
gives us an answer of 888.8 recurring. We need to round this answer to one
decimal place. The density of the cube is
therefore equal to 888.9.
As our units for mass were
kilograms and the units for the volume of the cube were cubic meters, then the units
for density will be kilograms per cubic meter. The density of a cube with side
length 0.15 meters and a mass of three kilograms is 888.9 kilograms per cubic
meter.
In our next question, we will
calculate the mass of a ball given its density and radius.
A steel ball has a radius of 10
centimeters. If the density of the ball is 8,000
kilograms per cubic meter, find the mass of the ball in kilograms. Give your answer to one decimal
place.
We are told that we have a ball
with radius 10 centimeters. The volume of any ball or sphere is
equal to four-thirds 𝜋𝑟 cubed. This means that we could calculate
the volume of the ball in cubic centimeters by substituting 𝑟 equals 10 into the
formula. As our units for density were
kilograms per cubic meter, we actually need to convert the radius into meters
first. As there are 100 centimeters in one
meter, 10 centimeters will be equal to 0.1 meters.
We can therefore calculate the
volume of the ball in cubic meters by multiplying four-thirds by 𝜋 by 0.1
cubed. Typing this into the calculator
gives us an answer of 0.004188 and so on. We were asked to calculate the mass
of the ball, which means this isn’t the final answer. We will therefore not round at this
stage.
We know that the density of any
object is equal to its mass divided by its volume. We can rearrange this formula so
that the mass is equal to the density multiplied by the volume. The mass of the ball will be equal
to 8,000 multiplied by 0.004188 and so on. This is equal to 33.5103 and so
on. Rounding to one decimal place, we
get an answer of 33.5. A steel ball with a radius of 10
centimeters, or 0.1 meters, and a density of 8,000 kilograms per cubic meter has a
mass of 33.5 kilograms.
In our next question, we will need
to calculate the volume given the density and mass of a block of aluminium.
The density of aluminium is 8,000
kilograms per cubic meter. Find the volume of a 100-kilogram
block of aluminium.
We know that the density of any
object is equal to its mass divided by its volume. The link between density, mass, and
volume can also be demonstrated in the triangle shown. In order to calculate the volume of
an object, we divide its mass by its density. If the mass is measured in
kilograms and the density in kilograms per cubic meter, then the units for volume
will be cubic meters.
In this question, the mass of the
block of aluminium is 100 kilograms and the density is 8,000 kilograms per meter
cubed. We can divide the numerator and
denominator by 100. This means that the volume of our
100-kilogram block of aluminium is one eightieth of a cubic meter.
In our last question, we will look
at how we can convert between different units of density.
The density of gold is 19,320
kilograms per cubic meter. What is this value in grams per
cubic centimeter?
In order to answer this question,
we need to work out how we can convert from kilograms per cubic meter to grams per
cubic centimeter. We know that there are 1,000 grams
in one kilogram. This means that in order to convert
from kilograms to grams, we need to multiply by 1,000. There are 100 centimeters in one
meter. In this question, we are dealing
with cubic meters, the units of volume. Cubing 100 gives us one
million. Therefore, there are one million
cubic centimeters in one cubic meter. This means that to convert from
cubic meters to cubic centimeters, we multiply by one million.
We can use this information to work
out how we would convert from kilograms per meter cubed to grams per centimeter
cubed. We need to multiply the numerator
by 1,000 and the denominator by one million. Let’s consider the fraction 1,000
over one million. We can divide the numerator and
denominator by 1,000, which gives us one over 1,000. This means that to convert from
kilograms per meter cubed to grams per centimeter cubed, we need to multiply by one
one thousandth. This is the same as dividing by
1,000.
We need to divide 19,320 by
1,000. Dividing by 1,000 moves all of our
digits three places to the right. This means that the density of
19,320 kilograms per cubic meter is the same as 19.32 grams per cubic
centimeter. The density of gold in grams per
cubic centimeter is 19.32.
To convert between the two
different units of density, we can divide or multiply by 1,000. To go from kilograms per cubic
meter to grams per cubic centimeter, we divide by 1,000. And to go the other way, we
multiply by 1,000.
We will now summarize the key
points from this video. We found out in this video that the
density of an object is equal to its mass divided by its volume. We also saw that we can rearrange
the formula density equals mass divided by volume to help us calculate either the
mass or the volume. Another way of demonstrating this
is using the triangle shown. The mass of an object is equal to
its density multiplied by its volume. And the volume is equal to the mass
divided by the density.
We saw that the standard units of
density are kilograms per cubic meter or grams per cubic centimeter. The units that the mass and volume
are measured in will dictate which units we’ll use for density.
In the final question, we saw that
we can convert between these two units of density by multiplying or dividing by
1,000. To convert from kilograms per cubic
meter to grams per cubic centimeter, we divide by 1,000. And to convert from grams per cubic
centimeter to kilograms per cubic meter, we multiply by 1,000.
