Andy French, Tony Ayres, Louisa Lintern Jones.

Department of Physics, Winchester College.

19/20 December 2016

The Cavendish Experiment

Measuring G using a torsional pendulum

Henry Cavendish

1731-1810

Isaac Newton

1642-1727

Isaac Newton proposed a Law of Universal Gravitation to calculate

the attractive force between point masses m and M, separated by

a distance r

11 3 -1 -26.67384 10 m kg sG

2

GMmF

r

The constant G is:

Henry Cavendish devised an ingenious

experiment to overcome this problem. His

equipment was based upon a torsional

pendulum. Amazingly, the tiny forces resulting

from the gravitational attraction of lead spheres

could result in a measurable effect upon the

oscillation of the pendulum.

To measure G experimentally on Earth is difficult, since

gravitational forces only become important on a human scale when

the masses are planet sized.

Measuring G via the Cavendish experiment

2 21 1

2 2

2 212

2

2 2

2

2

2

4

2

IT

I m L mL

T mL

mL

T

2

2

GmML

r

rG

LmM

Balance the torsion force (the twist) on

the wire with the torque resulting from the

gravitational attraction of masses M and m

2 2 2

2

2 2

2

2

2

mL rG

T LmM

LrG

MT

The torsion constant

can be found by

measuring the period

T of small oscillations

of the pendulum.

Moment of inertia

I of the pendulum

about the wire

axis

We can now combine the

expressions to find G

In the original

experiment

12

0.73kg, 158kg

1.8m, 230 4.1 mm

875.3s, 4.1mm

m M

L r

P L

2m

Tk

For a mass on a

spring of

stiffness k

Angle in

radians

Cavendish measured

11 3 -1 -2

11 3 -1 -2

modern

6.74 10 m kg s

[ 6.67 10 m kg s ]

G

G

At Winchester, we have a Cavendish Experiment kit supplied by LD Didactic GmbH

Red laser

Concrete

wall mount

Balancing

screws

Large lead

masses

1.496.6kgM

Small lead masses, fixed to torsional pendulum

Rotating mass

platform

Mirror

Torsion wire

0.015kgm

Diagram from LD Didactic Instructions P1.1.3.1

Laser Large mass

Torsional

pendulum & housing

Note the bracket can be mounted in a perpendicular

position – as in our case given we have the concrete

platform installed in Laboratory P4.

The laser is reflected off a mirror, which rotates with the pendulum. This causes

a vertical stripe to slowly track back and forth across the opposing wall of the

laboratory. The dynamics of these oscillations are used to determine G.

Tape measure scale

laser beam Torsional

pendulum

P4. Winchester College

Once the pendulum was aligned, measurements

were taken, every 30s, from the centre of the stripe

along the tape measure scale.

A measuring tape

was affixed to the

wall such that the

stripe tracks along it.

Vertical

laser

stripe

These screws are

used to lower

metal ‘baskets’

which lock the

pendulum

in place. The

baskets must be

raised when

the equipment is

transported. Horizontal alignment screws

Once the lowered baskets have released

the pendulum, it is often discovered that

the pendulum will ‘crash’ into the glass

windows. i.e. it has too much angular momentum and

the equilibrium position is not central. The top thumb

screw enables the wire and mirror to be manually

rotated to correct for this.

This process of alignment can be time consuming

and turns of the screw should be subtle!

This screw seems to relate

to how the wire is fixed in place.

Turning it has broken the

kit in the past....

‘baskets’

Don’t turn!

Diagrams from LD Didactic Instructions P1.1.3.1

The key feature

of the experiment

is to measure the

difference in equilibrium

positions of the laser spot

when the large mass

is in positions A (I)

or B (II).

The large mass

is rotated till it just touches

the glass panel

Wall

A laser range-finder was used to

determine the distances between the

equipment and the wall.

Errors of about 10mm (0.8%) due

to wobble of the laser when button

pressed, misalignment of walls etc

06820mmL

29mm

We actually measured

to a spot just above the

‘window’ rather than the

mirror, as the range-finder

tended to pick up the back

wall otherwise.

i.e. add 29/2 mm to

the range measurements.

The distance from the perpendicular

point N to the equilibrium of Position A

was measured to be

11292mmL

The offset angle of the laser will mean the

equilibrium positions are offset from N

Position A

equilibrium

point

Position B

equilibrium

point

06820mmL

3785mm 4032mm

5077mm

Tape

measure

Wall Perpendicular

position

11292mmL

From measurement, not inferred from data

Once the system was aligned,

and it was confirmed that

oscillations were contained

within the pendulum housing,

(i.e. no crashing against the

interior windows) laser stripe

positions along the tape were

recorded every 30s.

After four oscillations,

Position A was changed to

Position B, and another three

complete oscillations were

recorded.

This amounted to an

experimental run time of about

60 minutes.

Data was recorded directly into

Microsoft Excel and plotted as

the data was collected.

Note the equilibrium stripe

location for position A

will be inferred

via curve fitting (rather

than wait many hours!)

This could be a source of error

in both period and AB equilibrium offset

After several hours following the cessation of measurements recorded

in Position B, the following measurement was photographed.

This is assumed to be the Position B equilibrium reading:

4032mm

The data was then imported into MATLAB for further analysis.

1. Cubic spline interpolation to smooth between the data points.

2. Peak (and trough) finding routine to determine the peak and trough

times and positions for each A and B oscillation.

3. The difference between peak times were also used to calculate the

period of torsional oscillations, which was calculated to be 616s.

4. Curve fitting to estimate the longer term oscillation, and predict the final

equilibrium positions.

5. Calculation of G (see later slide for details)

6. Automatic graph plot and .png file creation.

Curve fit:

max( ) max

2( ) cost t

t tx t Ae x

T

Equilibrium

position

Curve fitting to determine the equilibrium position

max 0,t x

12max 1

,t T x

max 2,t T x

32max 3

,t T x

max 42 ,t T x

max( ) max

2( ) cost t

t tx t Ae x

T

12

32

0

1

2

3

T

T

T

x A x

x Ae x

x Ae x

x Ae x

12

32

12

12

0 1

2 3

0 1

2 3

0 1

2 3

1

1

1ln

T

TT

T

T

TT

x x A Ae

x x Ae Ae

x x ee

x x e e

x x

T x x

12

32

12

12

0 1

2 3

0 1

0 1

0

1

1

T

TT

T

T

x x A Ae

x x Ae Ae

x x A e

x xA

e

x x A

Use peak and trough coordinates

1.4966kg

32mm

0.015kg

6.9mm

M

R

m

r

Large sphere

Small sphere

R

50mm

46.5mm

d

b

Diagram and parameters from LD Didactic literature

The French

language version

(supplied with the kit!)

states 46.5mm whereas

the English language

version states 50mm....

We measured this

From LD Didactic

Instructions P1.1.3.1 pp3

Calculating G

2 2

2

1

1

, 2 ,4

1293tan

6820

1045tan

6820

A B

A B

A

B

r b L d

b dG

MT

A

B

2 2

2

2 LrG

MT

1.4966kg

50mm

46.5mm

1.083

M

d

b

K

2

2 3 3

1 1

2

11 3 -1 -2

46.5 10 50 10 1293 1045tan tan

1.4966 616 6820 6820

7.1 10 m kg s

G K

G

11 3 -1 -2

Cavendish

11 3 -1 -2

modern

6.74 10 m kg s

6.67 10 m kg s

G

G

So 6.8% out from the

correct value

Mirror rotation is twice pendulum

rotation, and between positions A

and B we double

Error analysis

1 1 1 11292 1046 1294 1044tan tan tan tan

6821 6819 6819 6821A B

1.49655 1.49665kg

49.5 50.5mm

46.45 46.55mm

1.083

615.5s 616.5s

M

d

b

K

T

22 3 3

1 1

max 2

11 3 -1 -2

max

22 3 3

1 1

min 2

46.55 10 50.5 10 1294 1044tan tan

1.49655 615.5 6819 6821

7.4 10 m kg s

46.45 10 49.5 10 1292 1046tan tan

1.49665 616.5 6821 6819

G K

G

G K

11 3 -1 -2

min7.0 10 m kg sG

2 2

2 A B

b dG

MT

There is probably

also a systematic

error in the larger

value due to the

inferred nature of

the equilibrium

position of A

We shall

assume a +/-

precision

error of 1mm

for all length

measurements

We’ll ignore

changes in K

So final result is: 11 3 -1 -2 11 3 -1 -27.0 10 m kg s 7.4 10 m kg sG

So we have about a 5% systematic offset

and a similar random error