Physics Lab Manual: Experiments on Motion, Forces, and Energy, Lecture notes of Physics

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Department of Physics and Astronomy

HERBERT LEHMAN COLLEGE

Laboratory Manual

Physics 166, 167, 168, 169

Writing a laboratory report OBJECTIVES The main way to communicate scientific information today is through articles and reports in scientific journals. Traditionally these were distributed in print, but can now be read in digital format as well, as shown in table 1. Table 1. Online resources for digitally distributed scientific publications Resource Topics URL arXiv Physics, Mathematics, Computer Science http://arxiv.org/ NASA Scientific and Technical Information (STI) Astronomy, Aerospace Engineering www.sti.nasa.gov/STI-public- homepage.html SOA/NASA Astrophysics Data System Astronomy, physics adswww.harvard.edu/ In college physics, you write a laboratory report for each experiment that contains the essential information about the experiment. For scientific information, a consistent format is helpful to the reader (and your lab instructor). Each laboratory report you turn in contains a subset of the sections found in a professional scientific publication for experimental topics, as shown in table 2. Table 2. A comparison of the sections of a laboratory report and a professional scientific publication Laboratory Report Professional Publication

1. Name, date and title of the experiment 1. Cover page: name, date, and title
2. Abstract 2. Abstract
   3. Introduction
   4. Methods and procedure
3. Data 5. Raw data and graphs
4. Calculations and analysis 6. Calculations and analysis
   7. Results
5. Conclusion 8. Discussion and conclusion The content to include in each section is detailed below. Your lab instructor requires all five sections to evaluate your work, so be sure to include every section in every report.

Introduction: Measurement and Uncertainty No physical measurement is ever completely precise. All measurements are subject to some uncertainty, and the determination of this uncertainty is an essential part of the analysis of the experiment. Experimental data include three components: 1) the value measured, 2) the uncertainty, and 3) the units. For example a possible result for measuring a length is 3. 6 ± 0. 2 𝑚. Here 3. 6 is the measured value, ± 0. 2 specifies the uncertainty, and 𝑚 gives the units (meters). ERRORS AND UNCERTAINTIES The accuracy of any measurement is limited by experimental errors and uncertainties. An error is a discrepancy between the measured value of some quantity and its true value. Errors in measurements arise from different sources: a) A common type of error is blunders due to carelessness in making a measurement, such as in an incorrect reading of an instrument. Of course these kinds of mistakes should be avoided. b) Errors also arise from defective or uncalibrated instruments. For example, if a balance does not read zero when there is no mass on it, then all of its readings will be in error, and we must either recalibrate it, or be careful to subtract the empty reading from all subsequent measurements. c) Even after we have made every effort to eliminate this kind of error, the accuracy of our measurements is still limited, due to experimental uncertainties. Uncertainties reflect unpredictable random variations in the measurement process: variations in the experimental system, in the measuring apparatus, and in our own perception! Since these variations are random, they will tend to cancel out if we average over a set of repeated measurements. To measure a quantity in the laboratory, one should repeat the measurement many times. The average of all the results is the best estimate of the value of the quantity. d) Besides the uncertainty introduced in a measurement due to random fluctuations, vibrations, etc., there is the so-called systematic or reading uncertainty which has to do with the limited accuracy of the measuring instruments we use. For example, if we use a meter stick to measure a length, we can, at best, estimate the length to about 1/10 of a millimeter. Beyond that we have no knowledge. It is important to realize that this kind of uncertainty persists, even if we obtain identical readings on repeated trials. To summarize, all measurements have an uncertainty (both random and systematic). Often we refer to this uncertainty as an error, but it should be emphasized that a true error reflects the difference between our results and the actual value of what we want to measure.

CALCULATING AVERAGES There are several important steps we will follow to help us quantify and control the errors and uncertainties in our laboratory measurements. Most importantly, in order to minimize the effect of random errors, one should always perform several independent measurements of the same quantity and take an average of all these readings. In taking the average the random fluctuations tend to cancel out. In fact, the larger the number of measurements taken, the more likely it is that random errors will cancel out. When we have a set of 𝑛 measurements 𝑥!, 𝑥!, ⋯ , 𝑥! of a quantity 𝑥, our best estimate for the value of 𝑥 is the average value 𝑥, is defined as follows. Average value: 𝑥 = !!!!!!⋯!!! !

The average value is also known as the mean value. Note that when making repeated measurements of a quantity, one should pay attention to the consistency of the results. If one of the numbers is substantially different from the others, it is likely that a blunder has been made, and this number should be excluded when analyzing the results. REPORTING ERRORS Quite often in these labs one has to compare a value obtained by measurement with a standard or generally accepted value. To quantify this one can compute the percent error , defined as follows. Percent error: % 𝐸𝑟𝑟𝑜𝑟 = !"#$!%# !"#$%! !""#$%#& !"#$% !""#$%#& !!"#$

× 100 (0.2)

Sometimes one has to report an error when the accepted value is zero. You'll encounter this situation in experiment 3. The procedure to follow is described at the end of that experiment. CALCULATING UNCERTAINTIES To estimate the uncertainty associated with our best estimate of 𝑥, we begin by examining scatter of the measurements about the mean 𝑥. Specifically, we start by determining the absolute value of the deviation of each measurement from the mean: Deviation: Δ𝑥! = 𝑥! − 𝑥 (0.3) Next we have to compare the deviation to the systematic or reading uncertainty due to limited accuracy of the instrument used. If this systematic uncertainty 𝑅 is bigger than the deviation Δ𝑥!, then the result of our measurements can be written as 𝑥 ± 𝑅 (0.4) If, however, the deviation is larger than the systematic or reading error, then we must determine how big the random uncertainty in our measurements is. This is given by the standard deviation, defined as follows.

SIGNIFICANT FIGURES A number expressing the result of a measurement, or of computations based on measurements, should be written with the proper number of significant figures , which just means the number of reliably known digits in a number. The number of significant figures is independent of the position of the decimal point, for example 2.163 cm, 21.63 mm and 0.02163 m all have the same number of significant figures (four). In doing calculations, all digits which are not significant can be dropped. (It is better to round off rather than truncate). A result obtained by multiplying or dividing two numbers has the same number of significant figures as the input number with the fewest significant figures. EXAMPLE Suppose that we want to calculate the area of a rectangular plate whose measured length is 11.3 cm and measured width is 6.8 cm. The area is found to be Area = 11. 3 𝑐𝑚 × 6. 8 𝑐𝑚 = 76. 84 𝑐𝑚! But since the width only has two significant figures we can round to two figures and report that the area is 77 𝑐𝑚!. GRAPHING, SLOPE AND INTERCEPTS In almost every laboratory exercise, you plot a graph based on the data measured or calculated. A graph lets you visualize the relation between two physical quantities. In plotting a graph, use the following steps:

1. Arrange the data into a table with two columns listing the values for the two measured or calculated quantities. For example, the first column could list the values for time and the second column could list the values for the average velocity.
2. Decide which of the two quantities to plot along each axis. Graphs have two perpendicular axes, the x-axis and the y-axis and by convention you plot the independent quantity along the x-axis and the dependent quantity along the y-axis.
3. Choose the scale for each axis to cover the range of variation of each quantity. You should choose the scale so that the final curve spans the largest area possible on the graph paper.
4. Label each axis with the quantity plotted on that axis and the units used.
5. Mark the main divisions along each axis.
6. Mark each data point on the graph using the values in each row of the data table. Data points must align with the value of each quantity on their respective axes. Make each data point clearly visible on the graph.
7. Fit and draw a smooth curve through the data points so that the curve comes as close as possible to most of data points. Do not force the curve to go exactly through all the points or through the origin of the coordinate system. The fact that not all points lie along the fitted curve just indicates that measurements are subject to some uncertainty. In many cases the fitted curve is a straight line. The best straight line fit has nearly the same number of data points above and below the line. The equation for a straight line is given by

Straight Line (0.8) The quantity b is the intercept: it is the value of y when x = 0. The quantity m is the slope of the curve. Given two points on the straight line, { , }, called basis points, the slope is defined as the ratio of the change in y to the change in x between these points, as shown in equation 0.9. Slope (0.9) Basis points are NOT experimental points. They should be chosen as far from each other as possible to increase the precision of m , as shown in figure 0.1. Figure 0.1 Choosing the correct basis points to calculate the slope y = mx + b y (^) 1 = mx 1 + b y (^) 2 = mx 2 + b 2 1 2 1

x x

y y

x

y

m

Experiment 1: Density OBJECTIVES Density describes how much matter is distributed within any given region of space. Quantitatively, it is the amount of mass contained in a unit of volume. In this experiment, you measure the mass and spatial dimensions of a specimen of an unknown metal, calculate its density, and use the result to identify the metal. The objectives of this experiment are as follows:

1. To identify the limits of precision for different measuring devices and calculate the uncertainty for measurements made with them
2. To measure the mass and spatial dimensions of a specimen
3. To calculate the volume and density of a metal specimen THEORY The density of a substance is constant for any pure sample at a constant temperature and pressure. For example, the density of a sample of pure Iron has the same density as any other sample of pure Iron at the same temperature and pressure. The density of any substance is its mass per unit volume, calculated using equation 1.1. Density ρ =

V

m

Where

• m = mass in grams (g), which you measure with a balance.
• V = volume in cubic centimeters (cm^3 ). The specimens in this lab are rectangular solids, so you calculate volume as the product of the specimen’s length, width, and height, each of which you measure using a different instrument.
• ρ = density in grams per cubic centimeter (g/cm^3 ), which you calculate using equation 1.1. ACCEPTED VALUES Metals in their solid state do not change significantly in density with minor changes in temperature and pressure. The accepted values for the density at 20˚C of a pure sample of the metals analyzed in this lab are as follows:
• Aluminum: 2.6989 g/cm^3
• Copper: 8.96 g/cm^3
• Iron: 7.874 g/cm^3 APPARATUS
• meter stick
• Vernier calipers
• micrometer calipers
• metal specimens
• platform balance

main scale. To take a measurement, place the specimen between the large jaws of the instrument. The hash mark on the main scale that aligns with the zero mark on the Vernier scale indicates the length of the specimen to within 1 millimeter (0.1 cm). The zero mark of the Vernier scale is the long line at the left end. Figure 1. 2 The Vernier calipers On the Vernier scale, the 10 divisions have the length of only 9 divisions on the main scale. Figure

1. 3 shows the relation between the two scales when the calipers are closed. Figure 1. 3 The initial positions of the main scale and Vernier scale The main scale is divided into 0.1 cm intervals and the Vernier scale is 0. 09 cm long, which provides an additional level of precision to one hundredth of a centimeter. In Figure 1. 4 , the zero line of the Vernier scale lies between the 2.3 cm and 2.4 cm marks of the main scale and the fourth mark on the Vernier scale aligns with a main scale line. Figure 1. 4 An example of the Vernier scale, which reads 0.04 cm past 2.3 cm or 2.34 cm

The difference between the size of a main scale division and a Vernier division is 0.01 cm. Therefore, the difference between the 2.3 main scale mark and the Vernier zero mark is 0.04 cm and the precise measurement is 2.34 cm. In general, if the n th line of the Vernier scale coincides with a main scale division, the Vernier zero mark is at a distance ( n × 0 .0 1 cm) beyond the main scale division immediately to the left of the Vernier zero mark. To find the length of the specimen, add this distance to the main scale measurement. For example, if line 1 on the Vernier scale coincides with a main scale line, the Vernier zero mark is 0.01 cm beyond the mark on the main scale. If line 2 on the Vernier scale coincides with a main scale line, the Vernier zero mark is 0.02 cm beyond the mark on the main scale, and so on. THE MICROMETER CALIPERS The micrometer calipers can measure distance to the nearest hundredth of a millimeter (0.001 cm). The construction of this instrument is shown in figure 1. 5. To measure a specimen, place it between the jaws (A and B). The spindle (B) moves by turning a precision screw connected to the thimble (D). Turning the thimble opens or closes the jaws. The distance between the jaws is given by the scale on the sleeve (C), which is ruled in millimeters. There are 50 divisions on the circular scale of the thimble. It takes two turns of the thimble to advance the spindle 1 mm, so each division on the circular scale of the thimble corresponds to an advancement of one hundredth of a millimeter. Figure 1. 5 The micrometer calipers To measure the length of a specimen, add the highest number of millimeters visible on the sleeve (C) to the hash mark on the thimble (D) that aligns with the horizontal axis on the sleeve. For example, if the edge of thimble (D) lies between 2.0 and 2.5 mm on the sleeve, as shown in figure

1. 5 , and the thirty-fifth mark of the circular scale is aligned with the horizontal mark of the sleeve scale, the distance between the jaws is 2.350 mm (0.2350 cm). If the thimble were rotated until the measurement on the circular scale was again 0.350 mm, then the sleeve scale measurement is between 2.5 mm 3.0 mm, and the total distance between the jaws is 2.850 mm (0.2850 cm). Before measuring lengths with either the Vernier calipers or the micrometer calipers, check the measurement when the jaws are completely closed to observe the systemic uncertainty caused by a consistent error in the measuring instruments. Record the measurement when the jaws are closed as the offset. If the offset is not zero, correct all other measurements by that amount. For example, if the micrometer measurement is 0.02 mm when the jaws are closed, then correct all measurements made by the micrometer by subtracting 0.02 mm from the micrometer measurement. This avoids a systematic error due to irregularities in the measuring instrument.

DATA

1. 1 Mass, Length, Width, and Height Measurement mass (g) length (cm) width (cm) height (cm) offset 1 2 3 CALCULATION AND ANALYSIS When analyzing repeated measurements, pay attention to the consistency of the results. If one of the numbers is substantially different from the others, it is likely that a procedural error has been made. Exclude a highly deviated value when analyzing the results.

1. Calculate the average measurements for the mass, length, width and height of your specimen.
2. Calculate the uncertainty of the average measurements for the mass, length, width and height of your specimen.
3. Calculate the volume of the specimen using the average measurements for length, width, and height.
4. How many significant figures does your result for the volume have?
5. Calculate the density of the specimen.
6. How many significant figures does your result for the density have?
7. Identify the metal from which your specimen is made using the standard density values from the theory section of this lab. Assume that the specimen is a pure metal.
8. Calculate the % error of the calculated density of the specimen using the standard density as the accepted value.
9. Suggest at least three hypotheses that could explain the % error and how you could change the experiment to eliminate that error.

Experiment 2: Acceleration of a Freely Falling Object OBJECTIVES Acceleration is the rate at which the velocity of an object changes over time. An object’s acceleration is the result of the sum of all the forces acting on the object, as described by Newton’s second law. Under ideal circumstances, gravity is the only force acting on a freely falling object. In this lab, you measure the displacement of a freely falling object, calculate the average velocity of a falling object at set time intervals, and calculate the object’s acceleration due to gravity. The objectives of this experiment are as follows:

1. To measure the displacement of a freely falling object
2. To test the hypothesis that the acceleration of a freely falling object is uniform
3. To calculate the uniform acceleration of a falling object due to gravity, g THEORY The average acceleration of an object measures how quickly its velocity is changing. If the velocity changes by an amount Δ v during a time Δ t , the average acceleration is calculated as shown in equation 2.1. Average Acceleration

t

v

a

In this experiment we study the motion of an object falling vertically down, that is, one-dimensional motion. Because the acceleration is constant, the average acceleration is equal to g. If the velocity of an object at t = 0 is v 0 , the velocity v at time t of an object moving in one dimension with constant acceleration g is shown in equation 2.2. Velocity v = v 0 + gt (2.2) If the position of an object at t = 0 is y 0 , then the position y at time t of an object moving in one dimension with constant acceleration g and an initial velocity of v 0 is shown in equation 2.3. Position 0 0 2 2 y = y + vt +^1 gt (2.3) Because the velocity of an accelerating object constantly changes, it is not possible to calculate the velocity at an exact time algebraically from measuring positions over time. However, you can approximate the velocity at the midpoint of any time interval by calculating the average velocity over

that time interval. The average velocity v of an object in one dimension as it moves along the y - axis

is defined as the change in its position, Δ y , over time, Δ t , as shown in equation 2.4. Average Velocity

t

y

v