Experiment: Oscillations of a Mass on a Spring

Transcription

1 Physics NYC F17 Objective: Theory: Experiment: Oscillations of a Mass on a Spring A: to verify Hooke s law for a spring and measure its elasticity constant. B: to check the relationship between the period and the amplitude. C: to find the relationship between the period and oscillating mass. D: To compare the estimation of the same parameter by two different methods. According to Hooke s law, the magnitude of extension (x) of a spring is proportional to the magnitude of tension in string (F el ): F el kx. (1) The constant k, is called the elasticity constant of spring or simply spring constant and has units of N/m. Theoretical calculations show that the oscillation period for a system constituted by a massless m spring with spring constant k and a block with mass m, is: T 2 (2) k A real spring, however, does have a certain mass m s and this mass participates in oscillations. To include the effect of this mass on oscillation parameters, one must refer to the effective m s mass me m instead of block mass m. For example, the expression for the period of 3 2 oscillations has to be written as : T m E (3) k Taking the square of both sides we get: T m E (4) k 2 4 So, the graph of T 2 vs m E must produce a straight line with slope = and intercept = 0. k Procedure: Part A: Hooke s Law 1) Suspend the spring from a clamp near the top of a retort stand and hang a small loop of string from the bottom of the spring. Clamp or tape a meter stick so that it stands vertically close to the spring but not touching it. Record the initial position L 0 of the bottom of the spring without any mass hanged. 2) Hang a 20 g mass on the loop of string at the bottom of the spring. With the mass stationary, measure the new position L 1 of the bottom of the spring. Subtract the initial position of the bottom of the spring to get the extension of the spring x 1 = L 1 - L 0 and record it in a table. Repeat for 30 g, 40g, (5 to 6 different masses up to 90 g) and find x 1 x 2 x Do not account spring mass in this experiment because its effect is included at initial extension. Calculate the absolute uncertainty Δx[cm]= ΔL +Δ L o = 2* ΔL (show why) 3) Convert all masses to weights(n), all extensions to meters and record them in a table. 4) Plot a graph of Force (the hanging weight) along the vertical axis vs extension along the horizontal axis. Don t forget error bars due to extension uncertainty. 5) Find the spring constant from the slope of the graph and report its estimation (k av +/- Δk ) to an appropriate number of significant figures. This is the elasticity coefficient measured by a static method.

2 2 Part B: Variation of Period with Amplitude 1) Measure the mass of the spring (m s ). 2) Add a string loop at spring end and suspend 50 g at the bottom of the string. (N.B. The string is there to prevent the mass from going into a twisting mode of oscillation.) Set the mass oscillating with the maximum possible amplitude. (It should not hit the table at the bottom nor go so high that the spring reaches its unstretched position). Measure and record the amplitude of the oscillations. 3) Measure the time for 5 complete oscillations. Start the stopwatch when the mass is at either its highest or lowest position. Wait a couple of cycles before starting so that you can start the watch at precisely the right time. Count zero when you start the watch and one and the end of the first oscillation, etc. Divide the total time by 5 to get the period of the oscillations and record it. Repeat this measurement at least 3 times. If the periods are very different, repeat again until you get three consistent results (similar values). When satisfied, calculate the average and the absolute uncertainty for the period estimation. If the three period readings have same value in seconds, use digits after the decimal point. 4) Repeat with the amplitude of the oscillations about 2/3 of what it was in the first case. 5) Repeat with the amplitude about 1/3 of what it was in the first case. 6) Record the measurements, the period and its uncertainty for each case in table 2. 7) Use the uncertainty bars and the method presented in Brief Review on Uncertainty to show that the period does not depend on the amplitude. Part C: Variation of Period with Mass 1) Using the same method as in Part B and a moderate amplitude, measure the period and calculate T Av and its uncertainty for each of masses: 10, 20, 30, 40, 50, 60, 70, 80 g. Calculate and enter the data for m E, T Av, ΔT in three rows of a table. 2) Calculate and Add in the table: - a row with T 2 Av value for each mass. - a row with T 2 uncertainty for each mass. 3) Plot a graph of T 2 vs m E. When plotting the graph T 2 vs m E, don t forget to draw the error bars. 4) Find the spring constant from the slope (Slope = 4 2 /k) of the graph T 2 vs m E and report its estimation (k av +/- Δk ) to an appropriate number of significant figures. This is the spring constant measured by a dynamic method. 5) Compare the estimations for spring constant found by the two methods (static and dynamic).

3 3 Conclusions: Part A: - Why one must not account the spring s mass in this experiment? - Draw an isolation diagram for the mass and explain how you get the value of F el. - Does the spring obey to Hooke s law? How do you know? What is its spring constant? Part B: - Do you expect the period to depend on the amplitude of the oscillations? - Does the experiment prove your expectations? Part C: - Is T 2 proportional to m E? How do you know? - Compare k-value s (from experiment A to that derived from the slope of the graph T 2 vs m E ) by using the overlapping of uncertainty intervals method. Do those values equal each other? Table No 1 L o =...[cm]; Δx = ΔL + ΔL o m[g] L[cm] x[cm]= L - L o Δx[cm] Ext.=X[m] ΔX[m] F el [N]=F G Table No 2 M=50g; T= t/5 [s] A[cm] A1= A2= A3= T[s] Tav[s] Δ T[s] Table No 3 m sp = [g] ; m E =m + m sp /3 ; T 2 = T 2 Av ; ΔT 2 = 2T Av ΔT m[g] m E [kg] T[s] T av [s] ΔT[s] T 2 [s 2 ] ΔT 2 [s]

4 VERTICAL SHM FOR A BLOCK-SPRING SYSTEM 4 When a block with mass m=1kg is hooked at the free end of a vertical spring it is extended by 30cm. Then, one pulls down the block and the system spring-block starts to oscillate. a) Show that, if the air friction is neglected these oscillations are a SHM b) Find their period. (1) (2) ( 3) F el No extension Δ Equilibrium Level Equilibrium y F G a_1) At state (2) the block is at equilibrium. So, F F F 0 We project this equation on Oy and get -kδ + mg = 0 and kδ = mg (1) a_2) At state (3) the block is moving and the second law of Newton is written NET We project on Oy and get k(δ+y) + mg = ma Y el G FNET Fel FG ma As -kδ - ky + mg = -mg -ky +mg = ma we get -ky = ma which can be written a = -(k/m)*y AND noting k/m = ω 2 2 d y 2 y (2) dt Eq. (2) is the SHM equation. b) The period of these oscillations is T 2 m / k. We can find k-value by using Hook s law for the initial extension. From (1) k = mg/δ =1*9.8/0.3 = 32.6N/m Then, T 2 1/ s Introduction to Excel: Using The Mass-Spring Experiment The purpose of this exercise is to learn many of the basic features of the spreadsheet program EXCEL. This type of program is most useful for doing repeated calculations such as one often encounters in analyzing the data from a physics experiment. It also has the capability of drawing graphs and other charts. Use your data from the mass-spring oscillations experiment. This exercise covers the analysis of the data from Part C: Variation of Period with Mass. Next, you can use what you have learned to set up a worksheet for Part A: Hooke s Law. Entering Data Start up EXCEL. Note that the screen is filled with a grid which will become the structure for a table of numbers. The columns of the table are labeled with letters and the rows with numbers. Each rectangle or cell thus has a unique identification such as A5 or M77. We will use the top row for the headings in our table. In the cells A1 to G1, enter the headings constants, mass, effective mass, period, "period squared ln m E and ln T. (Leave out the quotation marks.) You can move from one cell to the next using the arrow keys. For superscripts and

5 5 subscripts, select the required character(s). Under Format, select Cells and, in the Font dialog box that comes up, select Superscript (or Subscript). Click O.K. Since the cells are not wide enough for some headings, change the row width by clicking and dragging on the divider between the letters at the head of each column. In the second row, enter the appropriate units; e.g. (g) or (kg) or (s 2 ) etc. Leave the first column blank. In the cell A6 enter m s (g) and in A7 enter the mass of the spring in grams. In the second column, starting in cell B3, enter the hanging masses used in grams (one in each cell): 5, 10, 20 etc., 90. The last entry should be in B12. In the fourth column, starting in cell D3, enter the measured periods (one in each cell). The last entry should be in B12. Saving the Spreadsheet Save your work early and often under File, Save As. Give it a name that will make it easy to identify later. Editing Cell Contents To edit the contents of a cell, click on the cell. Its contents will appear in the box at the top of the chart. Edit the contents in the usual way in this space, then hit the enter key to make the changes in the cell or escape (Esc) if you do not want the changes. Creating Formulas The real power of a spreadsheet is in doing calculations. You do this by typing in a formula in one of the cells. The formula can contain numbers, mathematical expressions and the addresses of other cells in the spreadsheet. This allows the program to calculate quantities using the data you have entered. If you want to use the quantity in a specific cell, enter the cell address in the following format, called an absolute cell reference. For example, $A$3 refers to the contents of cell A3. If, on the other hand, you enter the address as simply A3, the spreadsheet uses it as a relative cell reference. This means, for example, that if you are entering the formula in cell C3 the calculation will use the contents of the cell two columns to the left of the current cell. A reference to C1 would call for the contents of the cell two rows above; and a reference to E5 would refer to the cell two columns to the right and two rows down. The advantage of this type of reference is that you can copy the formula into many other cells and the spreadsheet will automatically change the cell reference to the one that has the same location relative to the current cell. In EXCEL, all formulas start with an = sign. Note that this does not have the same meaning as it does in mathematics (two expressions being equal), but is just a signal to the program that you want it to do a calculation (as on a calculator). The standard operations are: +, -, * (for multiplication), / (for division) and ^ for raising to a power. The hierarchy for calculation is the same as for most calculators: multiplication and division first, then addition and subtraction. Use brackets in order to avoid any ambiguity. Other operations and functions can be found by clicking on the = sign outside and to the left of the space for editing cell contents and then on the down arrow to the left of it or in the Help menu under: Contents, Creating Formulas. Alternatively you can type in the function if you know the exact syntax. Example: to calculate the equivalent mass in cell C3, type =(B3+$A$7/3)/1000. This adds 1/3 of the mass of the spring ($A$7)to the hanging mass (B3) and converts to kg. When you hit enter the results of the calculation appear in the cell. Before you do anything else, use a calculator to check that the result of the calculation is correct. If it is not correct, check the formula and make any necessary changes. Similarly create formulas for T 2, Δ( T 2 ) in the appropriate columns. Copying Cell Contents You can copy the effective mass formula you have created in C3 into the rest of the cells in column C.

6 6 Click on cell C3. From the Edit menu, select Copy. Click and drag to highlight cells C4 down to C12. From the Edit menu, select Paste. Eureka! The effective mass is now calculated for all masses. Click on one of the cells containing the copied formula and note how the relative cell reference (B3) has been changed. Similarly copy the T 2, Δ( T 2 ) formulas to the rest of the cells in columns E, F and G. Displaying a Formula In all lab reports, it is necessary to show the formula being used to calculate results. One way to do this is to copy the formula into an unused cell in the column (e.g. copy cell C3 to cell C14) and then edit the formula to remove the = sign. Excel will then show the formula rather than calculated the result. Formatting the Cells The default formatting adopted by EXCEL usually doesn t look very good and almost always will not give the correct number of significant figures. For all lab reports, it is necessary to format the cells so that they show an appropriate number of decimal places. Click and drag to highlight all cells in a column; e.g. from C3 to C12. With the pointer anywhere in this area, click the right button on the mouse and select Format cells. Under the number tab, select number and set it for three decimal places (or whatever is appropriate). Click OK. Repeat for the other columns. Next, click and drag to select all cells in the table including the titles. Right-click in this area and select Format Cells, Alignment. Under Horizontal, select Center from the drop-down menu. Click OK. Drawing Lines around the cells Click and drag to select the entire work area from A1 to G12. Right-click in this area and select Format Cells. Under the Border tab, select the thin solid line under Style, then click on Outside and Inside. Click OK. Now select only the heading cells, A1 to G2. Right-click and select Format Cells, Border. Select the heavy solid line under Styles and click on Outside. Creating a Graph a) Create a graph of period vs effective mass, Click and drag to highlight the data in column C from C3 to C12. Hold down the Ctrl key and highlight the data in column D from D3 to D12. Click on the Chart Wizard icon in the tool bar. Under Chart Type, select XY scatter. Under Chart sub-type, select the scatter plot with no lines. Click Next and select the Series tab. Beside Name: type Period. Click Next and type Period vs Effective Mass under chart title. Type Effective Mass (kg) under Value (X) axis. Type Period (s) under Value (Y) axis. Under Gridlines, select major grid lens for the X axis and major grid lines for the Y axis. Click Next and select As object in Sheet 1. Click Finish. You will see your graph in the spreadsheet, probably covering some of the data. Click anywhere in the graph window outside the plot area and drag the graph (the pointer turns into crossed arrows) so that it doesn t hide any of the table of data. Click and drag on the little boxes at the corners of the graph box to make it larger. Note that all graphs in physics lab reports must be large enough to fill at least half a page. (If you want to change any of the options in the graph, right-click on what you want to change and select the appropriate changes.)

7 When you are finished, click anywhere outside of the graph box so that it is not selected. 7 b) Similarly, create a graph of period squared vs effective mass. Be sure to use the appropriate titles and units (e.g. s^2 for the Y axis). c) Size and position all graphs each one below the previous so that each occupies about a half a page. Adding a Regression (fit) Line The second and third graphs should be straight lines. To find the slope and intercept, you can instruct Excel to fit a straight line to the data points (a process called regression). Click anywhere in the chart. From the Chart menu at the top of the screen, select Add Trendline. Under Type, select Linear. Under Options, select Display equation on chart. If the equation is badly placed on the graph, click on it and drag it by one of the black squares to a more suitable location. If necessary, to remove a trendline, click on the line and select Clear. Printing Under the File menu, select Page Setup then select Header/ Footer, then Custom Header. Type a title for your table in the Left Section area and type your name in the Right Section area. Click OK. Next click on Print preview. If you are satisfied with what you see, click Print. If you want to change something, for example, the size or location of the graph, click Close, make the changes to the spreadsheet and select Print preview again from the File menu.