Introduction to lab measurements Purpose 1. To learn how to measure dimensions of an object using a Vernier caliper and a micrometer caliper. 2. To learn how to graph equations of a straight line and extract physical quantities. Introduction Part 1a: Making measurements using a Vernier caliper The Vernier caliper is an instrument used to measure external and internal dimensions of an object with precision up to one hundredth of a centimeter (or equivalently, one tenth of a millimeter). It uses a Vernier scale, pioneered by French mathematician Pierre Vernier in the 17th century. The purpose of the Vernier scale is to allow a good reading of a measurement that lies between two tick marks of a main scale (see numbered part 4 in figure 1). The way that this is done is by having a second, Vernier scale (see numbered part 6 in figure 1) that is used along with the main scale to provide additional accuracy. The graduations on the secondary scale have a different spacing to those on the main scale, such that N graduations on the secondary scale cover N-1 graduations on the main scale. Figure 1: a typical Vernier caliper. The numbered parts are: (1) The large, external jaws: used to measure external diameter or width of an object; (2) The small internal jaws: used to measure internal diameter of an object; (3) Depth probe: used to measure depths of an object or a hole (4) Main scale: this scale is marked every mm; (5) a Main scale: this scale is marked in inches and fractions; (6) a Vernier scale, which gives interpolated measurements to 0.1 mm or better; (7) a Vernier scale, which gives interpolated measurements in fractions of an inch; (8) Retainer: used to block movable part to allow the easy transferring of a measurement. Image and caption courtesy of Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Vernier_caliper.svg To be able to read the measurement using a Vernier caliper, first, look for where the zero on the secondary, Vernier scale lies against the main scale. In Figure 3(a), the zero on the secondary (Vernier) scale sits at zero on the main scale. This tells us that the reading is 0.0 cm. In Figure 3(b), the zero on the Vernier scale sits between 0.3 and 0.4 cm but we don't know just from whether the next decimal place is 0.31, 0.35, 0.37 cm, etc. For that second decimal place we look for where the secondary scale lines up with a division on the main scale. Again Figure 3(b), this occurs at the 8.5 mark on the Vernier scale. So the reading is 3.85 mm (or 0.385 cm). Try reading the Vernier scale on Figure 3(c) for yourself and see if you agree that the reading is 1.735 cm. The error in each reading is ±0.005 cm. For further examples of how to read the Vernier scale, and some practice where you can test yourself, see: http://www.technologystudent.com/equip1/vernier3.htm Brooklyn College 1 (a) (c) (b) Figure 2: Examples of Vernier caliper readings. In each picture, the main scale is on the body of the caliper while the lower scale is the Vernier scale, measuring to 0.005 cm. The readings are: (a) 0.00 cm; (b) 0.385 cm; (c) 1.735 cm (all ±0.005 cm) Part 1b: Making measurements using a micrometer caliper The micrometer works on a different basis, but can achieve even higher precision by interpretations between secondary scale marks. It is also supplied with a Vernier secondary scale (also known as a nonier scale) and so you can use a similar approach to make readings of the diameter of a steel ball at different locations. Examples of readings are given below, where interpretation between the micrometer scale divisions allows us to state a reading to the nearest 0.0001 cm, i.e. ± 0.0001 cm. (a) (b) Figure 3: Micrometer gauge readings. The micrometer scale on the screw can be interpreted to give readings of (a) 0.3651±0.0001 cm; (b) 1.0411±0.0001 cm. Brooklyn College 2 Part 2: Plotting theoretical physics’ equations in the form of a straight line Many of the experiments of the general physics 1 will require plotting the data in a particular form. When the requirement is to plot B versus A, then B is on the y-axis and A is on the x-axis. The y-axis is called the “ordinate” and the x-axis is called the “abscissa”. Usually the variables are set in the form of an equation of a straight line: where is the slope and c is the y-intercept (the value of when ). If the straight line passes through the origin point, then . Here is an example of Lab data: In Lab 2, you are asked to find by using your data for different values of and . In lab 2, you will study a relation of the form: g If you plot versus , you get a straight line. See figure 4. What is the slope of this straight line? And what is the y- intercept? Your answer for the slope should be g, and the y-intercept should be 0. The straight line passes through the origin. When we plot the graph, then we can use the value of the slope: Figure 4: Plotting data for a straight line, versus g to find the value of g, simply by multiplying the slope by 2. This shows a method that we use repeatedly in the general physics 1 lab. This method is: Extracting the desired quantity from the graph slope. In the above example, we have estimated from the graph slope that: g/2=512 cm/s2 by examining the trend line produced (in this case) by Microsoft Excel. (For more on the use of Excel, please see the Introduction to The Physics Laboratory document.) You may also extract the slope by simply calculating the gradient of the best fit straight line. So, if we now want to find the value of g, we just use this result to say that g = 2×512 cm/s2 =1025 m/s2. The actual number of significant figures used to express the result will depend on the accuracy of the measurements that led to the plot. For more on significant figures, see chapter 1 of the course textbook. In Part 2 of this lab, you will examine more examples of equation rearrangement to find out how the theoretical equation that describes an experiment can be used to construct a straight line graph from which we extract a slope value that tells us some particular physics. Running the experiment Note: the data sheet for this experiment is on page 5 Part 1: Making measurements using a Vernier caliper and a micrometer caliper a) Vernier Caliper 1) Open the simulator https://iwant2study.org/lookangejss/01_measurement/ejss_model_AAPTVernierCaliper/AAPTVernierCaliper_Simulatio n.xhtml Click the menu icon . Select ‘hint off’ and also select ‘answer off’ from the drop down menu of the hint show. Brooklyn College 3 2) Drag the lower scale of the Vernier Caliper to measure the length of the blue block. Record your reading. 3) Select ‘hint show’ and select ‘answer show’ from the hint drop menu. Compare your reading in step 2 to the answer displayed in yellow at the bottom right and also observe the hint displayed. 4) Change the length of the blue block to any other value and repeat steps 2 and 3. Take a screen shot only of this 2nd trial. b) Micrometer 1) Open the simulator https://iwant2study.org/lookangejss/01_measurement/ejss_model_Micrometer02/Micrometer02_Simulation.xhtml 2) Click “OK” in the message at the top. The message is: “drag the (Spindle or Ratchet) to measure length”. Now, drag the spindle to measure the length of the green block. While the micrometer is paused (do not move the mouse) and record your reading of the micrometer scale, and then compare it to the displayed reading in yellow background at the bottom right. Record the simulator displayed reading (that has a yellow background). 3) Click “Ok” in the message at the top “micrometer is paused…” and click the icon . Then drag the length of the green block to a larger value. Repeat the measurement for this second trial as in steps 2. Take a screen shot only of this 2nd trial. Part 2: Plotting theoretical physics’ equations in the form of a straight line A worked example If we have a theory that states that What would be the slope of the graph of versus ? For this, we have to rearrange the equation: So, if we plotted on the vertical axis and on the horizontal axis, we expect a straight line graph with slope (or "gradient") / . From that slope value we can find if we already have , or vice versa. Questions (Answering these questions provides further examples of extracting the theoretical slopes of straight line graphs). All the theories for these equations will be studied latter in your physics course. So don’t worry about their definitions now. a) If we have a theory that states that What would be the slope of a graph of vs ? And what units would the slope have, if is a time measured in seconds, s and is a mass measured in kilograms, kg? b) If we have a theory that states that What would be the slope of a graph of vs in this case? ( is called the stiffness.