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Physics Part IA Cambridge University, 2012-2013

Notes for Experimental Physics Dimensional analysis: The dimension of a quantity specifies the nature of that quantity. It can be presented in two forms - specific and abstract:

* Specific dimensions: Commonly referred to as units. The standard specific dimensions of mass, length and time are kg, m and s, respectively.

* Abstract dimensions: These do not specify the particular unit in which a value is given, only its fundamental nature. The abstract dimensions of mass, length and time are expressed as [M], [L] and [G], respectively. Dimensional analysis: Since dimensions specify the nature of a quantity, it is possible to deduce how given quantities are related to other quantities by comparing their dimensions. This is called dimensional analysis: Example: Determining how the period

T

of a simple pendulum depends on mass

m

l :

and length

T

m is [M] and l is [L]

*

Abstract dimension of

*

This necessitates the presence of another, constant quantity in the expression which

is [T],

incorporates units of time - this is the gravitational constant,

g , which has

abstract dimension [L][T]-2

T

Expressing

a b

T [?]m l g

generally in terms of

m ,l

and

g :

g

a

-2 g

b

[?] [ T ] =[ M ] [ L ] ( [ L ][ T ]

) = [ M ] a [ L ] b +g [ T ]-2g

[?] a=0,-2g=1 [?] g =

1 -1 1 1

[?] b- =0 [?] b=

2 2 2

-1

[?]T [?]l2 g 2

[?]T [?]

[?]

l g

Limitations of dimensional analysis:

* Dimensional analysis can obviously only be applied to quantities which have dimensions

* It therefore cannot give the constant of proportionality in an equation, since such constants are dimensionless numbers

* The relationship between quantities obtained via dimensional analysis will not be unique if some or all of the constituent quantities can be combined to give a dimensionless quantity. Such a dimensionless quantity could be multiplied into the

Physics Part IA Cambridge University, 2012-2013 equation or excluded without affecting the overall dimension, and hence there exist an infinite number of possibilities for the form of such an equation

Experimental error and uncertainty: The final result of any experiment is always expressed in the form is the value of the result obtained and

[?]A

A +- [?] A , where

A

is its associated uncertainty. Uncertainty is

caused by error, which is intrinsic to every experimental procedure, and can be one of two types - random or systematic.

Physics Part IA Cambridge University, 2012-2013 Types of error: Random error: The effect of natural fluctuations in measured values from the central/mean value:

* Equally likely to be positive or negative

* Can be detected by analysing the spread of data

* Can be reduced by repeating measurements to reduce the spread of the data

* Generally affects the precision of the result Systematic error: The effect of an experimental arrangement which differs from that assumed:

* Not equally likely to be positive or negative

* May vary or remain constant throughout the experiment

* Not a result of the natural spread of data, hence cannot be reduced by repeating measurements

* If its source and size are identified, it can be removed from the final result

* Systematic error generally affects the accuracy of the result Distribution of experimental data: By the Central Limit Theorem, experimental data collected with a large number of trials is always normally-distributed, thus follows a Gaussian distribution of the form: 2

1 f ( x )=

e

[?] 2p s 2 Where

-( x-m ) 2 2s

m is the mean of the distribution and s

is its standard deviation. This leads to

the following properties of experimental data:

*

The mean obtained from the data distribution

*

m , such that

x is an approximation of the true mean of the

x [?] m

m , thus

The maximum value of the distribution is at the true value represented by the data

x x

is the best indication of

*

Data are therefore always represented by

*

Experimental uncertainty is the uncertainty in this mean value,

for all trials conducted

s mean

Result=x +- s mean Where

s mean

x and

s mean

are always given to the same number of decimal places, and

is given to two significant figures if it begins with a 1 or a 2, and one significant

figure otherwise. Obtaining the value of

x Since experimental data

X by:

from experimental data: usually have discrete values

x , the mean is simply given

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