Macroscopic and Microscopic Springs

Transcription

1 Macroscopic and Microscopic Springs PY205m 1 Purpose In this lab you will: investigate the spring-like properties of a straight wire, discover the stretchiness of a material, independent of the size and shape of an object, relate macroscopic stretchiness of a material to microscopic stiffness of the interatomic bonds. Some useful equations and constants are given at the end of this document. 2 Questions to Explore 1. Is a solid metal straight wire like a spring? If so, in what ways? If you consider the wire to be a spring, how does spring stiffness compare to the coiled springs used in the previous lab? 2. How can one express the spring-like properties of the particular metal in a way that is independent of the size, shape, and physical dimensions of the wire (Young s modulus)? 3. How can one relate the macroscopic spring-like properties of the particular metal to the stiffness of that metals interatomic bonds (interatomic spring stiffness). 3 Background 3.1 Spring forces The magnitude of the force exerted by a spring on an object attached to it is linearly proportional to the absolute value of the stretch of the spring. The spring constant k s, represents the stiffness of a particular spring, and has units of N/m. The stretch can be positive or negative, and is defined as the difference between the current length of the spring L and its original, relaxed length L 0. Sometimes the symbol L is used to represent the stretch: L = L L 0. Since we will be dealing with both a macroscopic and a microscopic stretch in this lab, we will refer to the macroscopic stretch as L. This makes the force law for a spring: 3.2 Young s modulus F = k s L The stiffness of a wire depends on the length of the wire and the thickness of the wire, so different wires made from the same metal will have different stiffnesses. It is useful to have a measure of the stiffness of a particular material (such as aluminum, copper, gold, carbon nanotubes), independent of the dimensions of the wire. 1

2 Calculating Young s modulus is a way to measure the springiness of a material and factor out the size and shape of the particular wire. Young s modulus is the ratio of: Force exerted per square meter of cross-sectional area (F/A) ( stress ) to Fractional stretch of the wire ( L/L 0 ) ( strain ) 3.3 Interatomic spring stiffness Y = (F/A) ( L/L 0 ) Our simple model of a solid object is a bunch of tiny balls (atoms) that are held together by springs (chemical bonds). The relaxed length of the little spring between two atoms (the interatomic bond) is just the distance from the center of one atom to the center of the other atom, d, which for our model, is just twice the radius of the one of the atoms since the electron cloud of the atom fills in the extra space. A simple cubic arrangement of the atoms would mean that the volume that each atom fills would be d x d x d and would have a cross-sectional area of d x d. Since the interatomic bonds are modeled as springs, they also have a stiffness k s,i that relates the interatomic force to how much those interatomic bonds stretch. We will use s to refer to the microscopic stretch. 4 Experimental Setup F = k s,i s Equipment: a straight metal wire stretched on a frame, with a hanger for weights and a millimeter scale for determining the position of the end of the wire, Vernier caliper (see 4.2 for how to read). 4.1 Recording data From the WebAssign, right click on the Excel file youngs modulus.xls, and save it to My Documents. Double-click the file to open it in Excel. You should see a display that includes places for your group information, measured values, and calculated values, as well as a space for your graph. You will record all of your data and analysis on the Excel sheet and and submit it to WebAssign to be graded by your TA. 4.2 Measuring with Vernier Calipers In this laboratory exercise you will be using a Vernier caliper, which is comprised of two teeth, one attached to a fixed scale and the other attached to a sliding (Vernier) scale. In order to measure an object s width, the object is simply placed between the caliper s two teeth. The sliding tooth is then moved until the object is pressed tightly between the teeth. Using both scales, the width can be read to the nearest cm (or 0.05 mm). The scales are read as follows (refer to the figure): 2

3 1. Find where the 0 mark of the sliding scale lines up on the fixed scale. In this case, it is just past the 5.3 cm mark. So, the first reading is 5.3 cm. 2. Find the mark on the sliding scale that most closely lines up with one of the marks on the fixed scale. Here, 5.0 and 6.0 are very close, but 5.5 lines up best with one of the marks on the fixed scale. This value is the number of hundredths of centimeters (or tenths of millimeters). So, the second reading is cm. 3. Add the two values together to get the total reading: 5.3 cm cm = cm To get more practice reading calipers, visit the following website which has an interactive Java applet: Note: At this website, the label on the sliding scale is misleading. The scale is labeled mm, which leads the user to think that the numbers on that scale represent millimeters. This is NOT correct. As described above, the numbers on the sliding scale represent tenths of millimeters. 5 Measurements 5.1 Dimensions of the wire One of the goals is to describe the stretchiness of a material independent of the size and shape of the wire. To do this, we first need to know the physical dimensions of the wire. Measure the following and enter your data into your Excel sheet: Entire length of the wire without any weights on it. Diameter of the wire using the caliper (see section 4.2) Using the measured diameter of the wire, calculate the cross-sectional area of the wire. CHECKPOINT: Compare your data with another group to make sure it is reasonable. 3

4 5.2 Stiffness of the wire To determine if the straight wire behaves like a spring, you need to measure the position of the end of the wire as you add different amounts of weight. Remove the kinks in the wire: If the wire has kinks in it, you will measure the springiness of the coiled wire instead of the stiffness of a straight wire. Remove the kinks before starting your measurements. Add 6 kg to the platform hanging from your wire. Remove the 6 kg. Repeat. This helps get small kinks out of the wire, so the straight wire doesnt act like a coiled spring. Take the following measurements and record your data in your Excel sheet: Look at the scale on the apparatus, and make sure you know what the divisions are, and how to read the scale. You should be able to measure to the nearest 0.1 mm. You need to make three sets of measurements, so that you can get average values. Take measurements with 0 kg, then 2 kg, 4kg, 6kg. Take all the masses off and start from zero again, taking the same measurements twice more. Take the average of your three sets of measurements. Calculate the variation of your measurement: The variation in the measurements (the reproducibility ) gives you an indication of how meaningful the individual measurements are. Record your estimate of the variation. If the measurements are 5.5, 5.7, 5.4, 5.6, 5.8, and 5.6 mm, the average is ( )/6 = 5.6, from a minimum of 5.4 to a maximum of 5.8. A compact way to report this average with the approximate variation is mm. If the variation is larger than 0.3 mm, you MUST repeat your measurements. Discard earlier measurements since this was probably getting more kinks out of the wire. BEFORE graphing your data, discuss the following questions with your group members: If the wire acts like a spring, what would you expect a graph of stretch versus force to look like? How could you determine the spring stiffness of the wire from the graph? Use your data to make a graph in Excel of the stretch of the wire (on the y-axis) versus the force exerted on the wire (on the x-axis): Both of these quantities will need to be computed from your data (either by hand or using Excel). From the two columns representing stretch and force, highlight the cells with the data for 2, 4, 6 kg. Note: Do NOT include the data for 0 kg, because the wire may be kinked without a load. Choose menu option Insert Chart, and choose XY (Scatter). (Do NOT choose a Line chart). Be sure to verify that you have the proper quantities on the proper axes. Fit your data to a Trendline : Right click on one of the data points and select Add Trendline 4

5 Choose Linear under Type Under Options, choose Display Equation on Chart this will display the equation of the line that best fits your data in the form y = mx + b. You can reposition the equation. From your data, determine the spring stiffness of the wire. Since F = k s s, then s = 1 k s F, so the slope of your graph is equal to 1/k s. CHECKPOINT: Compare your measurements, calculations, and graph with another group to verify that they are reasonable. 5.3 Stretchiness of the metal Calculating Young s modulus is a way to measure the springiness or stretchiness of the metal and factor out the size and shape of the particular wire. BEFORE: calculating Young s modulus, discuss the following questions with your lab partners and write your answers on a whiteboard: If the cross-sectional area of the wire were doubled, would the wire stretch more, less, or the same amount when you hung a 2 kg mass on it? Why? Think about putting two of your original wires together side by side. How does this compare to placing two springs in parallel? If the wire were twice as long, would it stretch more, less, or the same amount when you hung a 2 kg mass on it? Why? Think about connecting two of your original wires end-to-end. How does this compare to placing two springs in series? The quantity F/A is called stress. What are its units? The fractional stretch L/L 0 is called strain. What are its units? CHECKPOINT: Compare your responses with another group. Use the measurements for your wire to determine Young s modulus: At equilibrium F = k s L = s, so: Y = (F/A) ( L/L 0 ) = (k s L/A) ( L/L 0 ) = k sl 0 A Use your measurement for the initial length, your calculated value for the cross-sectional area, and the value for the spring stiffness obtained from your graph to determine Young s modulus for the material of which your wire is made. Show your calculation on your whiteboard and record your result on your Excel sheet. How does your value for Young s modulus compare to that of aluminum ( N/m 2 ) or lead ( N/m 2 )? Given that the wire is brass (an alloy of copper and zinc), is your value reasonable? CHECKPOINT: Compare your results with another group. 5

6 5.4 Determining the interatomic spring stiffness Young s modulus is a way to relate the a macroscopic spring-like properties of the particular metal to the stiffness of that metal s interatomic bonds (interatomic spring stiffness). BEFORE: determining the interatomic spring stiffness, discuss the following with your partners: Assuming a simple ball and spring model for our solid, how could we determine the length of the interatomic bonds if we knew the density of the material? (Hint: You have probably done something similar in your homework recently) Using this idea and the following information, determine the length of the interatomic bond for brass? Brass is an alloy of copper and zinc. The atomic mass of copper is 64 grams/mole; zinc is 65 grams/mole. The density of copper is 8.94 grams/cm 3, for zinc, it is 7.14 grams/cm 3. Note: beware of your units Use your calculation of the interatomic bond length and your value for Young s modulus to approximate the stiffness of the interatomic bonds (k s,i ): In Section 5.3, we showed: Y = k sl 0 A For our ball and spring model, the cross-sectional area of the bond is A = d 2 and the relaxed length is L 0 = d (if you are unsure why this is, reread Section 3.3). Thus: Y = k sl 0 A = k s,id d 2 = k s,i d Show your calculation on your whiteboard and record your result on your Excel sheet. CHECKPOINT: Compare your results with another group. Then submit your Excel sheet and answer the follow-up questions in WebAssign. September 25,

7 Things you must know: Definition of average velocity Definition of momentum Definitions of particle energy, kinetic energy, and work The Momentum Principle The Energy Principle Other fundamental principles: L dl A A = τ net,a t or = τ net,a dt L trans,a = r A p (point particle) τ A = r A F r cm = m 1 r 1 + m 2 r m 1 + m Multiparticle systems: P tot M tot v cm (v << c) K tot = K trans + K rel K rel = K rot + K vib K trans 1 2 M totv 2 cm (v << c) I = m 1r m 2r K rot = L2 rot 2I = 1 2 Iω2 L trans,a = r cm,a P tot Lrot = I ω LA = L trans,a + L rot γ Other physical quantities: 1 ( ) E 2 (pc) 2 = ( mc 2) 2 v 2 1 c F grav = G m 1m 2 r 2 ˆr U grav = G m 1m 2 r F grav mg near Earth s surface Ugrav mg y near Earth s surface F elec = 1 q 1 q 2 4πǫ 0 r 2 ˆr U elec = 1 q 1 q 2 4πǫ 0 r F spring = ks s opposite to the stretch U spring = 1 2 k ss 2 for ideal spring U i 1 2 k sis 2 E M approx. interatomic pot. energy E thermal = mc T E N = 13.6eV N 2 where N = 1,2,3... (Hydrogen atom energy levels) ksi E N = N ω 0 + E 0 where N = 0,1,2... and ω 0 = (Quantized oscillator energy levels) m a ( ) d p dt = pv R mv2 (if v << c) where R = radius of kissing circle R ω = 2π T Y = F/A L/L (macro) x = Acos ωt ω = Y = k si d (micro) ks m speed of sound v = d ksi m a 7

8 ˆf = cos θ x,cos θ y,cos θ z unit vector from angles Moment of intertia for rotation about indicated axis L L R I = 2 5 MR2 I = 1 2 MR2 I = 1 12 ML2 I = 1 12 ML MR2 R Ω = (q + N 1)! q!(n 1)! S k ln Ω 1 T S E S = Q T (small Q) prob(e) Ω (E) e E kt Constant Symbol Approximate Value Speed of light c m/s Gravitational constant G N m 2 /kg 2 Approx. grav field near Earth s surface g 9.8 N/kg Electron mass m e kg Proton mass m p kg Neutron mass m n kg Electric constant 1 4πǫ N m 2 /C 2 Proton charge e C Electron volt 1 ev J Avogadro s number N A atoms/mol Plank s constant h joule second hbar = h joule second 2π specific heat capacity of water C 4.2 J/kg Boltzmann constant k J/K milli m kilo K micro µ mega M nano n giga G pico p tera T