Physics Lab Report Analysis Of A Bubble Chamber Picture
CR680610
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 𝑥 = !! ! =
17.
Next find (𝑥! − 𝑥)!
Then 𝑠 = (!!!!)!
! !!! !!!
= 2.21
To calculating standard uncertainty u when a set of several repeated readings has been taken, use
𝑢 = 𝑠 𝑛
(The standard uncertainty of the mean has historically also been called the standard deviation of the mean, or the standard error of the mean.)
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:
𝑝! = 1 2 (𝑝!! + 𝑝!!)
Where 𝑝!! 𝑎𝑛𝑑 𝑝!! are initial and final momentums of Σ particle. And can be found using the measured track length 𝑙!. The length of time that the Σ−lives(the time between its creation and decay) is
𝑡 = 𝑙! 𝑣 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑎𝑡 𝑟𝑒𝑠𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑏
𝑡 = 𝑙! 𝑣
1 − 𝑣!
𝑐! 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑚𝑜𝑣𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
𝑡 = 𝑚!𝑙! 𝑝!𝑐