Introduction to Laboratory Work - Mechanics Lab | PHYS 1210, Lab Reports of Physics

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PHYS 1210 / 1310 – Mechanics Lab Lab schedule 1210 sum09 T May 19th 10:30-12 Lab intro and lab 1: Using motion sensors to determine motion W May 20th 10:45-12 lab 2: Projectile Motion R May 21st 10:45-12 lab 3: Forces T May 26th 10:45-12 lab 4: Work and Energy W May 27th 10:45-12 lab 5: 1dim- collisions R May 28th 10:45-12 lab 6: Conservation Laws T June 2nd 10:45-12 lab 7: Intro to Rotational Motion W June 3rd 10:45-12 lab 8: Ramp Race- Moment of Inertia and Motion R June 4th 10:45-12 lab 9: Pendulum T June 9th 10:45-12 lab 10: Damped and Driven Pendulums, Resonance W June 10th 10:45-12 lab 11: Standing Waves on a Rope, Wave Velocity R June 11th 10:45-12 lab 12: Speed of Sound: Kundt Tube and Water Tube Experiment ‘0’ Introduction to laboratory work Lab Safety: General lab conduct The general rules of conduct for laboratory work as displayed on the laboratories news boards apply. It is good practice to learn from the beginning a lab conduct which does not encourage dangerous habits: Use safety equipment where advised. Clean up trash before you leave. Switch appliances to ‘off’ or ‘standby’. Do not leave an experiment behind in an unknown or unsafe state, eg hot plates must be switched off and hot water must be emptied into the sinks. The most important rule for laboratory safety is: Be prepared and do not behave in a careless manner. Put equipment back in the place where you originally found it and take special care to not leave behind any unidentifiable substances or spills – unknown substances present a safety concern for anyone who has to use the workspace after you and not everyone who may work in these labs may use only harmless substances. Notify your TA about broken equipment! General safety equipment First Aid Kit Room 130A at door Emergency eye flash Room 130A at door Fire Extinguishers at least one per lab, near exits Special advice for Phys 1210 lab In the Mechanics and Thermodynamics Lab, room PS133, there are no special regulations regarding electrical or chemical exposure. You are advised to be cautious of electrical lines and other electrical equipment or chemicals which might be misplaced in the room. Otherwise, the electricity which is used is comparable to that of normal household exposure. Watch out for broken glass or swinging objects. Measurement One can only measure quantities. Measuring a quantity can be achieved by direct or indirect means using tools and apparatuses or the senses. To measure a quantity means to determine its ratio to the unit employed in expressing the value of that quantity. This definition must be clearly distinguished from the process that is used in the measurement process. Example: We could measure the weight of a metal cylinder by placing it on a scale or by immersing it in water. Either of these processes is a true measurement of the body’s weight but neither is a direct measurement. Very few measurements are direct. Commonly it is only done directly for the determination of length. Example: Angles are, for instance, not measured by use of a wedge-like unit-degree but by taking a length on a curved linear scale. Temperature is measured as a length on the stem of a thermometer resulting from the expansion of a liquid (mercury). The degree of precision with which an observer can read a given linear scale depends upon the definiteness of the marks on the scale and the skill with which the observer can estimate fractional parts of scale division. In many high precision instruments, the linear scale is provided with some sort of vernier, a mechanical substitute for the estimation of fractional parts of scale divisions. Its use requires skill and judgment. Quality instrumentation usually carries a symbolic code printed on the front or back which indicates how to use the device. The device manual explains the symbols. Common advice is to keep an instrument level and operate it in certain temperature ranges. The Impossibility of Exact Measurements: An absolutely exact measurements is impossible. Example: If we weigh a small piece of material on a precision balance, a typical result could be 1.7438 grams. This is only an approximation to the true weight, just as the value 3.1416 is only an approximation to the number . A more sensitive balance would give a more accurate number. This is true for all measurements. A common way of increasing the accuracy of a measurement result is to repeat a measurement many times and to build averages and measures of data scatter. Error of Measurement The hindrance in obtaining the correct values by measurement is known as experimental error. Example: Suppose the piece of material to be weighed again, by the same person and on the same balance. The result will almost certainly turn out to be different by some small amount. This means simply that neither result is correct. Even if we double our efforts to repeat the first measurement as exact as we can, the new result will differ and in fact it might now differ more from the first result than the first repetition did. The causes of error in precise measurements are many and various. A systematic study of errors gives rise to a mathematical analysis, based on the principles of probability and is known as The Theory of Errors or Error Theory. As a first categorization, we distinguish accidental (random) and systematic errors. By using precise instruments, the accuracy of the value we extract can be increased. It is our task to determine the ‘most accurate’ value of a quantity and to work out its actual accuracy. The difference between the observed value of any physical property and the (unknown) accurate value is called the error of observation. Random Errors are disordered in their incidence and variable in their magnitude, changing from positive to negative values in no ascertainable sequence. They are usually due to the observer and are revealed by repeated observations. In physics, we encounter a number of phenomena and instrument types, which add a random error known as ‘noise’. Noise is not at all due to the observer but is due to fluctuation of properties on an atomic level (not necessarily fluctuation of the quantity of interest itself but of the directly measured quantity). Systematic errors may arise from the observer or the instrument. They are usually the more troublesome, for repeated measurements do not necessarily reveal them and even when known they can be difficult to eliminate. Unlike accidental errors, systematic errors may shift the observed value away from the actual value, in other words they can add an offset to the measurements. If a quantity x0 is measured and recorded as x we shall call x-x0 the error e in x0. x= x0+e where e can be positive or negative. e/x0 is known as fractional error. 100e/x0 is called the percentage error. If only a single measurement is made any estimate of the error may be widely wrong. To obviate or reveal random errors, repeated measurements of the same quantity are made by the same observer with the same experimental setup under identical environmental conditions. Example: A sequence of values for the acceleration by gravity on earth may look like this: 9.78, 9.81, 9.81, 9.79, 12.5, 9.80 [m/s2]. It seems quite possible in this particular data set that some mistake was made in recording ’12.5’ and it is reasonable to exclude that value from further analysis. Note, however, that there is no definite line from where on such a judgment should be made. Thus, results should not be rejected indiscriminately or without due thought and should remain in the actual experiment’s lab-book. Abnormal or unusual results, followed up rather than discarded, have often led to important discoveries, both, on the very small (our 2310 lab) as well the very large (noble prize awards) scale. The above measurements indicate that the measured quantity lies between 9.78 and 9.81 m/s2, and as the arithmetic mean of the five measurements is 9.798, we can say that the value of the measured quantity is 9.80 +-0.02 to indicate the scatter of the data about the mean. In well defined frequency curves (good statistics), Xmod is the x value where the peak occurs, Xmed is shifted a bit to the side which is the more prominent branch, and XAM is furthest out on that side. In physics, in the absence of systematic errors, we often obtain symmetric data distributions (‘normal distribution’). In that case all three average values are identical. The Geometric Mean G is the n-th root of the product of all data values. Example: data 2, 4, 8 - G = (2x4x8)1/3 = 4 The Harmonic Mean H is the number of all data points divided by the sum of the inverse of all data values. Example: data 2, 4, 8 - H = 3/(7/8) = 3.43 In general, H < = G < = XAM (read ‘less or equal’) The Root Mean Square or quadratic mean is: rms = (x2/N)0.5 Example: data 1, 3, 4, 5, 7 - rms = {(1+9+16+25+49)/5}0.5 = (20)0.5 = 4.47 Quartiles (Q1 to Q3), Deciles (D1 to D9), and Percentiles (P1 to P99) extend the idea of the median to those values which divide a set of data into 4, 10, and 100 equal parts, respectively. They are used to express where a given data point lies in the distribution. Measures of Dispersion of Data Range, mean deviation, standard deviation, variance The range of a set of numbers is the difference between the largest and smallest numbers in the set. Mean or average deviation MD: The sum of the absolute of the differences between all data points and XAM divided by the number of data points. Example: data 2, 3, 6, 8, 11 - MD = (/-4/+/-3/+/0/+/2/+/5/) /5 = 2.8 Standard Deviation s is the average of the square of the difference between data and XAM: {(x-XAM)2/N}0.5 Example: data 2, 3, 6, 8, 11 - s = ((16+9+0+4+25)/5)0.5 = 3.29 Variance  is the square of s. For normal distributions 68.27% of cases are between XAM –s and XAM +s. 95.45% of data lie within 2s of XAM and 99.73% are within 3s. For two sets of numbers N1 and N2 with variances s12 and s22, the combined variance is s2 = (N1s12 + N2s22)/(N1+N2) For moderately skewed distributions MD = 4/5 s Relative dispersion: absolute dispersion / average [%] Error Bars Once the standard deviation has been determined, it can be included in the graphical analysis of the data. 0 2 4 0 10 20 data points: black squares error bars: indicated about data time [s] po si tio n [m ] Here, an error bar has been determined for the position. Note, that there can be cases where one has an error bar for the other dimension too (here time).