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Mass
Volume
and
Density
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Contents
1
Volume .................................................................................................................................................................. 3
1.1
Calculating volume. .................................................................................................................................... 3
1.2
Volume 2 ...................................................................................................................................................... 5
1.3
Symbols and conversion. .......................................................................................................................... 5
2
Mass...................................................................................................................................................................... 6
3
Density ................................................................................................................................................................. 7
3.1
The ratio table ........................................................................................................................................... 7
3.2
Proportional................................................................................................................................................. 7
3.3
Calculating density .................................................................................................................................... 7
4
Practice exercises VMBO - t. ......................................................................................................................... 8
5
Practice exercise HAVO/ATHENEUM ......................................................................................................12
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Density is a substance characteristic. It is the number of grams of 1 cm³
substance. In the chapter below we are going to look at the meaning of:
 volume;
 mass;
 density.
1 Volume
Volume is the space taken up by substance.
Volume can be calculated for a regular solid.
With an irregular solid, the volume can be determined by using the immersion
method.
1.1 Calculating volume.
SOLID
CUBE
CONTENT
Surface  6  a  a  6a 2
Content  a  a  a  a 3
RECTANGULAR
BLOCK
Surface  2  l  b  2  l  h  2  b  h
Content  l  b  h
CYLINDER
Surface  2    r 2  2    r  h
Content    r 2  h 
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1
4
  d2  h
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SOLID
CONTENT
PYRAMID
Surface  Surface  area  faces
l
Content 
CONE
1
3
 Surface  area  h
Surface   r (r 2  h 2 )   r l
Content 
PRISM
1
3
  r 2  h
l b
h
2
SPHERE
Surface  4    r 2    d 2
Content  4 3    r 3 
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1
6
  d 3
=
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1.2 Volume 2
With an irregular solid, the volume can be determined by using the immersion
method.
𝑉𝑠𝑡𝑜𝑛𝑒 = 𝑉𝑒𝑛𝑑 − 𝑉𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔
1.3 Symbols and conversion.
Quantity
V
1 mL
=
1 cm³
1L
=
1 dm³
Unit
cm³
1L = 10 dL = 100 cL = 1000 mL
1 m³ = 1,000 dm³ = 1,000,000 cm³ = 1,000,000,000 mm³
1 m³ = 1 x 103 dm³ = 1 x 106 cm³ = 1 x 109 mm³
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2 Mass
A substance consists of atoms. The mass (the number of grams) of an atom depends on the
core particles.
The mass of a substance is the same everywhere. This doesn’t change.
A cube of 100 grams on earth has also 100 grams of mass on the moon.
The number of atoms is ultimately still the same.
The term weight doesn’t count here because weightlessness doesn’t change the number of
molecules.
When we talk about grams or kilograms in physics, then the quantity is mass.
Quantity
mass
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m
unit
gram g
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3 Density
Density is the mass of one cubic centimetre of a substance
If the volume and the mass are known, then the mass of 1 cm³ substance can be calculated.
3.1 The ratio table
Mass in grams
Volume in cm³
1
3.2 Proportional
If the mass is twice as much, then the volume is also twice as much
Mass : volume = constant
Graphically this is a straight line through the origin.
3.3 Calculating density
Volume
V
cm³
Mass
m
g
Density
ρ
g/cm³
𝜌=
𝑚
𝑉
Example:
m = 250 g
V = 50 cm³
If there are two substances with different
densities, the substance with the lowest
density will float.
At the same density, the substance
mixes/floats. Think of a fish/submarine.
ρ = ? g/cm³
𝜌=
𝑚
𝑉
Never take the average density of a
mixture.
𝜌=
250𝑔
50𝑐𝑚3
Add up the volume and add up the mass.
Calculate the density.
𝜌 = 5 𝑔/𝑐𝑚³
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4 Practice exercises VMBO - t.
1) Erica wants to investigate whether iron is heavier or lighter than aluminium.
She can do this by weighing pieces of iron and aluminium that:
a. are the same size.
b. are equally heavy.
2) By density is meant:
a. The volume of one gram of a substance.
b. The volume of 1 cm3 substance.
c. The mass of 1 g substance.
d. The mass of 1 cm3 substance.
3) Marijke says: “You can recognise a substance by it’s density.”
Willem says: “Density is a substance characteristic.”
Who is right?
a. Both.
b. Neither.
c. Marijke.
d. Willem.
4) What is the formula for calculating density?
𝑣𝑜𝑙𝑢𝑚𝑒
a. 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑚𝑎𝑠𝑠
𝑚𝑎𝑠𝑠
b. 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑉𝑜𝑙𝑢𝑚𝑒
c. 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑚𝑎𝑠𝑠 × 𝑉𝑜𝑙𝑢𝑚𝑒
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
d. 𝑀𝑎𝑠𝑠
=
𝑉𝑜𝑙𝑢𝑚𝑒
𝜌=
𝜌=
𝑉
𝑚
𝑚
𝑉
𝜌= 𝑚×𝑉
𝑚=
𝜌
𝑉
5) Fill in :
a. You measure the density of a substance in ………………per ………….. and you write it as
……/……
b. Aluminium has a
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smaller/equally large/larger
density than steel.
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6) Density is a substance characteristic. This doesn’t change. You can look up the density in a
table. You can find density in table 15 of the Binas.
a. Complete the table. Take care that the
Metal
metals are given in order of density.
(You can recognise the metals in the Binas
Density
3
in g/cm
Aluminium
because the specific resistance in the last
Order
number
2.7
column is smaller than one)
b. What is the lightest metal?
_________________________
c. What is the heaviest metal?
_________________________
7) You can calculate density using
…………. Divided by ……………
8) In the pictures below there are three objects, A, B
and C
Calculate the density of the objects.
𝑚𝑎𝑠𝑠
𝑚
6,0 𝑔
= _____𝑔/𝑐𝑚3
A.
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
B.
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
C.
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙 × 𝑏 × ℎ = ____𝑐𝑚 × ___𝑐𝑚 × ___𝑐𝑚 = _____𝑐𝑚3
𝑚𝑎𝑠𝑠
𝑚
𝑔
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
𝜌=
=
= ______𝑔/𝑐𝑚3
𝑉𝑜𝑙𝑢𝑚𝑒
𝑉
𝑐𝑚3
A mass = 6.0 g
Volume = 5.0 cm3
𝑉𝑜𝑙𝑢𝑚𝑒
𝑚𝑎𝑠𝑠
𝑉𝑜𝑙𝑢𝑚𝑒
𝜌=
𝜌=
𝑉
𝑚
𝑉
=
=
5,0 𝑐𝑚3
𝑔
𝑐𝑚3
= ______𝑔/𝑐𝑚3
B mass = 27 g
C mass = 80 g
Volume = 10 cm3
Volume = __ cm3
h=4.0 cm
l=1.5 cm
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b=1.5 cm
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9a
Calculate the density of the substances in figure 9
𝑚𝑎𝑠𝑠
𝑚
_________
= __________ = _____𝑔/𝑐𝑚3
A.
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
B.
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑒𝑛𝑑 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 = … … . . − ⋯ … . . = ⋯ … . . 𝑐𝑚3
𝑉𝑜𝑙𝑢𝑚𝑒
𝜌=
𝑉
𝑚
_________
=
= _____𝑔/𝑐𝑚3
𝑉 __________
C. 𝑉𝑜𝑙𝑢𝑚𝑒 = ⋯ … . . − ⋯ … . . = ⋯ … . . 𝑐𝑚3
𝜌=
𝜌=
𝑚
_________
=
= _____𝑔/𝑐𝑚3
𝑉 __________
D mass = 26 g
Volume = 10 cm3
E mass
= 68 g
F mass
= 63 g
initial position = 53 cm3
initial position = 41 cm3
end position
= ____
end position
= _____
Volume
=
Volume
=
b) Write down for each object what kind of substance it could have been made of. Search in
the Binas from witch materials the objects are made
Object D is made of _________________________
Object E is made of _________________________
Object F is made of _________________________
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10 With a graduated cylinder and scales (and a
little calculating) you can determine the
density of a liquid (figure 10).
a. Read off how great the volume and the
mass of the liquid are:
volume = ________mL
mass = ________g
b. Using these details, calculate the density
of the liquid.
____________________________________
____________________________________
c. What liquid could this be? Use table 15 of your Binas
____________________________________
11
The blocks A and B are made of different substances.
Which substance has the greatest density?
_______________________________
How can you tell that?
_________________________
Two blocks keep each other in balance.
Extra The density of water and ice
12
An ice block is 3.0 cm long, 2.0 cm wide and 1.5 cm high.
The mass of the ice block is 8.1 g.
a. Calculate the density of the ice with these details ______________
b. Does the outcome make sense with what it says in the handbook?
Explain your answer.
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5 Practice exercise HAVO/ATHENEUM
1HA About density
a
Explain why we call density a substance characteristic.
b
Give the formula of density.
2HA Density in another unit
Janna has a chunk of nickel with a volume of 1.3dm3 and a mass of 11.6 kg.
a
How many cm3 is equal to 1.3 dm3?
b
How many g is equal to 11.6 kg?
c
Calculate the density of nickel in g/cm3.
Now calculate in the same way the density of the three following substances in
g/cm3:
d
e
f
Magnetic steel, from which there is a horseshoe magnet that has a volume of 135
cm3 and a mass of 0.93 kg.
A wooden chess piece with a volume of 1458 mm3 and a mass of 1250 mg.
Fuel oil with a mass of 0.151 tonnes standing in a barrel of 159 litres.
(1 tonne = 1000kg)
3HA How many types of wood?
Marnix has a number of wooden toys. He wants to investigate how many different
types of wood the toys are made of. To do that he determines the mass and volume
for each toy. This gives the following results:
Volume (cm3)
12
21
16
31
25
28
14
22
11
a
mass (g)
7.3
12.8
12.8
18.9
15.2
14.4
8.5
17.6
6.7
Put these values into a diagram, with volume on the x-axis and mass on the y-axis.
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Using the diagram, say how many types of wood have been used for Marnix’s toys.
4HA The water in a measuring glass rises, like in the image
on the right, when you drop the tin horse into it.
Calculate the mass of the tin horse.
5HA The wooden block here on the right has a mass of 160 g.
Calculate the density of the wood from which the block
has been made.
6HA Tjeerd has determined the mass and volume of five
brass objects. He has collected the results of his
measurements in the table below.
a)
Put Tjeerd’s measuring results into a graph.
b)
Read off from this graph how great the mass is of a brass object with the
volume of 5.0 cm 3 .
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7HA Jantje has a piece of wood in his hands.
If he drops the piece of wood into a graduated cylinder with water in, it sinks.
The level in the graduated cylinder rises by 15 ml to 85 mL.
a)
Calculate the initial position of the water before the wood was placed in it.
b)
What type of wood was the block made out of?
c)
Calculate the mass of the block of wood.
8HA How heavy is sugar water on the stomach?
a
If you dissolve a little sugar in water, the volume of the mixture (sugar water) seems
to be just as great as the volume of the water. Carla mixes 60.0 cm3 water with 4.0g
sugar.
Explain how it is possible that the volume of the fluid doesn’t change, even though a
substance has been dissolved into it.
Carla has to calculate the density of the sugar water. For this she makes first of all
two drawings: one drawing of the situation before the dissolving and one from the
situation after the dissolving. She puts the details in both drawings and afterwards
calculates the density as required.
B HA Produce the same drawings as Carla.
C A Calculate the density of the sugar water.
9A
a
b
c
Oil mixture
John has 10 dm3 petrol and wants to mix oil through it.
He has to mix at 1 to 25, that means one part oil to 25 parts petrol.
The oil has a density of 0.80g/cm3, the petrol 0.75 g/cm3.
How many cm3 of oil does John need?
Give a reasoned estimate for the density of the petrol-oil mixture.
Calculate the density via the mass of the petrol — mass of the oil — total volume.
John uses so little oil that when mixing it the volume of the petrol doesn’t change.
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10A On top of a well there is a round iron lid with a surface area of 0.385 m² and a
thickness of 3.0 cm.
Calculate the mass of the lid.
11A
There are 32 l petrol in a can. The whole weighs 23.8 kg.
Calculate the mass of the empty can.
12A A measuring glass has a mass of 235 g.
Calculate the total mass of the measuring glass if this is filled with milk up to the
measuring line of 85 ml.
The density of the milk is 1.028 g/cm³
13A A cylindrical measuring glass has a diameter of 6.0 cm.
It is filled with 0.55 kg of mercury.
Calculate how high the liquid comes to in the measuring glass.
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