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Understanding Mass, Weight, and Density: Early Measurements and Newton's Contribution, Exercises of Acting

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.

What you will learn

  • How was the density of Schiehallion mountain used to estimate the Earth's density?
  • What was the significance of Cavendish's experiment in determining the Earth's density?
  • What role did Isaac Newton play in estimating planetary densities?
  • What were the early attempts to measure the Earth's density?
  • How does mass differ from weight?

Typology: Exercises

2021/2022

Uploaded on 09/27/2022

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HOW MUCH DOES THE
EARTH WEIGH?
To be strictly pedantic, a better question
is
WHAT IS THE MASS OF THE
EARTH?
!Mass, weight and density.
!The role of Isaac Newton (ca. 1687).
!Early attempts to determine the density of the
Earth:
• Pierre Bouguer (1749),
• Nevil Maskelyne (1774).
!Henry Cavendish and the Cavendish
experiment (1797-98).
!Later attempts.
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Download Understanding Mass, Weight, and Density: Early Measurements and Newton's Contribution and more Exercises Acting in PDF only on Docsity!

HOW MUCH DOES THE

EARTH WEIGH?

To be strictly pedantic, a better question

is

WHAT IS THE MASS OF THE

EARTH?

! Mass, weight and density.

! The role of Isaac Newton ( ca. 1687).

! Early attempts to determine the density of the

Earth:

  • Pierre Bouguer (1749),
  • Nevil Maskelyne (1774).

! 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

as the density of soil is about 1! 2 " that of

water and typically rocks are 2! 3 " times that

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

  1. 000000002 times,

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.

Model of the apparatus used by Henry Cavendish

to measure the relative density of the Earth. He

started his experiments in 1797.

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

!!

Model to the apparatus used by Johann Wilsing

to measure the relative density of the Earth. His

experiments commenced in 1887.

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.