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The concepts of mass, weight, and density, focusing on early attempts to measure the Earth's density by Pierre Bouguer and Nevil Maskelyne. The text also discusses the differences between mass and weight, and how Newton's Law of Gravitation was used to estimate planetary densities.
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! Mass, weight and density.
! The role of Isaac Newton ( ca. 1687).
! Early attempts to determine the density of the
Earth:
! Henry Cavendish and the Cavendish
experiment (1797-98).
! Later attempts.
Some differences between mass and weight:
! Mass is related to the amount of matter or
substance contained in an object. It determines
how easy or difficult it is to “move” an object
! Weight is the force by which an object is
attracted to the Earth (or some other body, like
another planet or the Moon).
m 1
m 2
Some differences between mass and weight:
! Mass is an intrinsic property of an object as
it remains the same everywhere in the universe.
! Weight is variable depending on the
gravitational force acting on the object.
If your bathroom scale reads 120 lb on Earth, it
would read
25.5 lb on the Moon
58.1 lb on Mars
140 lb on Saturn
364 lb on Jupiter
Some differences between mass and weight:
! Mass can be measured using a pan or lever
balance comparing the unknown mass with a
known mass. The measured mass will be the same
everywhere.
! Weight is measured using a spring balance.
But, the actual reading would depend on location.
Another quantity we will come across is:
! Density. Is physical property of matter, as
each element and compound has a unique density.
Density =
Mass of an object
Volume of the object
! Mass of an object
= density " volume of the object.
Often, the density of a substance is compared with
the density of water (the relative density).
Kennedy half-dollar
(copper-nickel)
mass # 11 $ 34 g
gold
24 $ 46 g
lead
14 $ 37 g
tin
9 $ 26 g
ice
1 $ 16 g
Examples of densities relative to water:
Substance Relative density
Cork ~0.
Pine ~0.
Hickory 0.
Tin 7.
Iron 7.
Nickel 8.
Copper 8.
Silver 10.
Lead 11.
Uranium 19.
Gold 19.
Platinum 21.
Iridium 22.
Osmium 22.
Clearly, we cannot measure the mass of the Earth
directly. Even though the volume of the Earth
was known, its density could only be guessed at
of water. So, what to do?
Newton considered the possibility of measuring
the attraction between two objects directly. He
imagined two solid spheres, each 1 foot in
diameter, made of the same material as the Earth.
He stated that if they were spaced
apart, they
would not
“ even in spaces void of resistance come
together by the force of their mutual
attraction in less than a month’s time. ”
He went on to say,
“ Nay, whole mountains will not be sufficient
to produce a sensible [measurable] effect. ”
Newton had made an error leading to an
enormous underestimate of the attractive force.
In fact, the two solid spheres would come into
contact in about 5
minutes instead of a month!
Had he realized his mistakes, he would not have
been so quick to dismiss the possibility of
measuring the attraction. Fortunately, not
everyone agreed with him at the time!
But, he had recognized that there were two
possible methods,
! using the deflection of a plumb-line by a
mountain, and
! by measuring the force between two
spheres directly.
The gravitational force of attraction between two
people, each of 150 pounds sitting 1 8! !apart, is
approximately equivalent to the weight of a
single grain of sand.
Which means the force is about
or two-billionths, the weight of each person!
One consequence of Newton’s Law:
A line drawn through freely hanging, stationary
pendulum, or plumb-line, on a smooth, spherical
Earth would always pass through the center of
the Earth and point upwards towards the zenith.
!
zenith zenith
zenith
center of the Earth
Schiehallion (3,550 ft), a mountain located in
central Scotland near Loch Rannoch, was selected
as a good candidate for deflection experiments. It
is almost conical in shape making a determination
of its volume a relatively straightforward task.
In 1774, Nevil Maskelyne began making
measurements of the deflection of a plumb-line on
the north and south sides of Schiehallion. He
showed that a plumb-line was indeed deflected
(by about 0. 0032
!
) but it took a further two years
for the mountain to be surveyed to provide its
volume and mass.
Dr. Nevil Maskelyne FRS (1732-1811)
A detailed survey of Schiehallion was carried out
by Charles Hutton. He estimated its density was
about 2.5 times that of water, which meant the
Earth was about 4.5 times as dense as water. In
1821, following a more detailed study, he revised
his result upwards to 4.95. Using this latter result,
the mass of the Earth would be,
5 , 230 , 000 , 000 , 000 , 000 , 000 , 000 , 000 kg,
i.e., 5.23 septillion kilograms!
Charles Hutton (1737-1823) depicted on a bronze
medal in 1821 (National Portrait Gallery, London).
Sketch of Henry Cavendish (1731-1810), dressed
in a favorite, old-fashioned frock coat and three-
cornered hat. Cavendish refused to sit for a
portrait; this sketch was drawn surreptitiously by
William Alexander at a Royal Society dinner ( ca
1800). It is held in the British Museum.
Cavendish’s house on the southside of Clapham
Common, in southwest London. He lived there
from 1782-1810 and it is where he carried out his
historic experiment to determine the density of
the Earth. The house was demolished in 1905.
Cavendish carried out his experiments inside a
1 0 !" 1 0! brick outbuilding at his house in
Clapham, London; he operated the equipment
from the outside. He observed the deflection of
the torsion pendulum using telescopes at each
end of the building and moved the large masses
into position by a system of pulleys.
He made a series of 17 separate experiments
altogether, between August 5th, 1797 and May
23rd, 1798, comprising 29 measurements of the
density. He concluded the density of the Earth
was
times that of water.
Assuming the Earth is a sphere with an average
radius of 6371 km, then its mass is
5 , 901 , 000 , 000 , 000 , 000 , 000 , 000 , 000 kg,
which is within 1.2% of today’s accepted value.
His experiment, which involved the elimination
of many spurious effects, has been heralded as
one of the greatest experiments in the history of
science.
In 1879, Phillipp von Jolly carried out a series of
experiments using a modified common beam
balance and four glass globes identical in size -
two containing mercury and two evacuated - and
a large lead ball, weighing 5,772 kg (about 6
tons). He found the average density of the Earth
was 5. 692 greater than that of water.
21 m
0 .5686 m
!!
Key:
C - Cavendish (1798)
J & C - James and Clarke (1856)
v J - von Jolly (1881)
W - Wilsing (1887)
Today’s value
C
J & C
v J
W
A comparison of the densities of the planets
(relative to water) in the solar system.
Planet Relative density
Mercury 5.
Venus 5.
Earth 5.
Mars 3.
Jupiter 1.
Saturn 0.
Uranus 1.
Neptune 1.
