Physics Lab Report Analysis Of A Bubble Chamber Picture

Measurement of the angle θ

For better understanding I am showing you a different particle track diagram bellow. Where at

point C particle 𝜋 − 𝑎𝑛𝑑 Σ− are created and the Σ− decays into 𝜋 ∓ 𝑎𝑛𝑑 K− particles

The angle θ between the π− and Σ− momentum vectors can be determined by drawing tangents

to the π− and Σ− tracks at the point of the Σ− decay. We can then measure the angle between

the tangents using a protractor. Alternative method which does not require a protractor is also

possible. Let AC and BC be the tangents to the π− and Σ− tracks respectively. Drop a

perpendicular (AB) and measure the distances AB and BC. The ratio AB/BC gives the tangent of

the angle180◦−θ. It should be noted that only some of the time will the angle θ exceed 90◦ as

shown here.

Determining the uncertainty of Measurements

In part B, It is asked to estimate the uncertainty of your measurements of 𝜃 and r.

Uncertainty of measurement is the doubt that exists about the result of any measurement. You

might think that well-made rulers, clocks and thermometers should be trustworthy, and give

the right answers. But for every measurement - even the most careful - there is always a margin

of doubt.

It is important not to confuse the terms ‘error’ and ‘uncertainty’.

Error is the difference between the measured value and the ‘true value’ of the thing being

measured.

Uncertainty is a quantification of the doubt about the measurement result

Since there is always a margin of doubt about any measurement, we need to ask ‘How big is the

margin?’ and ‘How bad is the doubt?’ Thus, two numbers are really needed in order to quantify

an uncertainty. One is the width of the margin, or interval. The other is a confidence level, and

states how sure we are that the ‘true value’ is within that margin.

You can increase the amount of information you get from your measurements by taking a

number of readings and carrying out some basic statistical calculations. The two most

important statistical calculations are to find the average or arithmetic mean, and the standard

deviation for a set of numbers.

The ‘true’ value for the standard deviation can only be found from a very large (infinite) set of

readings. From a moderate number of values, only an estimate of the standard deviation can be

found. The symbol s is usually used for the estimated standard deviation.

Suppose you have a set of n readings. Start by finding the average:

For the set of readings x={16, 19, 18, 16, 17, 19, 20, 15, 17 and 13}, the average is �̅� = ∑ 𝑥𝑖

𝑛 =

17.

Next find (𝑥𝑖 − �̅�) 2

Then 𝑠 = √ (𝑥𝑖−�̅�)

2

𝑛−1 = 2.21

Lifetime calculation

In part C you are asked to determine the life time of the neutral particles from their

momentums.

The Σ− lifetime can be approximately determined using the measured values of the Σ− track

lengths. The average momentum of the Σ− particle can be found from its initial and final values:

𝑝Σ̅̅ ̅ = 1

2 (𝑝Σi + 𝑝Σf)

Where 𝑝Σi 𝑎𝑛𝑑 𝑝Σf are initial and final momentums of Σ particle. And can be found using the

measured track length 𝑙𝛴 . The length of time that the Σ−lives(the time between its creation and

decay) is

𝑡 = 𝑙𝛴 𝑣

𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑎𝑡 𝑟𝑒𝑠𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑏

𝑡 = 𝑙𝛴 𝑣

(1 − 𝑣2

𝑐 2 ) 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑚𝑜𝑣𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

𝑡 = 𝑚0𝑙Σ 𝑝Σ𝑐