W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 12 ~ A" A Aa Ap Find Normal Title No Spacing Heading : Heading 2 Paste Replace BI U~ ab X x A LA Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor S1b: Pendulum Harmonic Motion and Hooke's Law Introduction: This experiment examines Hooke's law and Simple Harmonic Motion. Simple Harmonic Motion (SHM) is when the position of a body can be described as a sinusoidal function of time. This motion is over the same path, which includes an equilibrium position that the simple harmonic oscillator must pass through with each cycle. Examples of SHM include the oscillation of a mass attached to an ideal spring (which will be studied in this experiment), the motion of a simple pendulum (also studied in this experiment) within a grandfather clock, the ticking of a metronome, and many others. SHM occurs whenever the net force or the net torque acting on a body is directly proportional to the displacement of the body from its equilibrium position. An example of such force, which will be studied in this lab, is the restoring force exerted by a stretched or a compressed spring, described by Hooke's law. Another example is the simple pendulum which can be described ideally as a point mass suspended by a massless string from some point about which it is allowed to swing back and forth in a place at a small angle (0 String >Metal sphere >Protractor Meter Stick Digital Caliper String >Spring Masses (10 fifty-gram Photogate OneDrive . . . X masses), Meter stick > Support rods Screenshot saved Computer timing system & photogate The screenshot was added to your Painter Hooked Bet of masses OneDrive. Figure 1 Page 1 of 10 1954 words X Text Predictions: On x Accessibility: Investigate " Focus 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 12 ~ A" A Aa Ap Find Normal Title No Spacing Heading : Heading 2 Paste Replace BIUvab x x'A LAB= = ~ ~ Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor Theoretical background: Simple harmonic motion is characterized by a number of parameters. Amplitude, A, is the maximum deviation of an object from its equilibrium position. Period, 7, is the time it takes to complete one full cycle of the motion. Frequency, f, is the number of oscillations of the object per second. Angular frequency, @, is related to the period as w = The time that it takes to make one complete oscillation is defined as the period T. We also use frequency to describe the periodic motion of simple harmonic oscillation. The frequency f is the number of oscillations that occur per unit time and is the inverse of the period. f= 1/T (1) Similarly, the period is the inverse of the frequency, T=1/f (2) Oscillation of a Pendulum When a simple pendulum is displaced from its equilibrium position, there will be a restoring force that moves the pendulum back towards its equilibrium position. If the restoring force F is opposite and directly proportional to the displacement x from the equilibrium, so that it satisfies the relationship F= -kx (3) then the motion of the pendulum will exhibit simple harmonic motion and its period can be calculated using the equation for simple harmonic motion. T = 2n (4) It can be shown that if the amplitude of the motion is kept small, equation (4) will be satisfied. OneDrive . . . X Screenshot saved The screenshot was added to your OneDrive. N Date Modified 03/29/22 Page 2 of 10 1954 words X Text Predictions: On x Accessibility: Investigate " Focus 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap Find Normal Title No Spacing Heading : Heading 2 Paste Replace BI Uvab X x A LA Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor L F= -mg sine mg Figure 2. Diagram of forces for a Pendulum. The component of the gravitational force along the arc provides the restoring force F and is given by F = -mg sin 0 (5) For amplitudes where the angle (0) is small, sin 0 can be approximated by 0 measured in radians so that equation (3) can be written as F = -mg 0 (6) The angle 0 in radians is (), the arc length divided by the length of the pendulum (the radius of the circle in which the mass move). The restoring force is then given by F= -mg () (7) Page 3 of 10 1954 words X Text Predictions: On x Accessibility: Investigate " Focus + 74%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap Find Normal Paste Title BIUvab X X A LA No Spacing Heading : Heading 2 Replace Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor According to equation (7), this restoring force is directly proportional to the displacement x and satisfies the SHM relationship (equation (3)), where k = -9. Substituting k into equation (4), the period of a simple pendulum can be found by T = 21 (8) Which simplifies to T = 2nt SIFI (9) Therefore, for small amplitudes the period of a simple pendulum depends only on its length and the value of acceleration due to gravity. In order to get the gravitational acceleration g we will square both sides of equation (9) (10) Now, if we construct a graph of T2 as a function of L we can calculate the gravitational acceleration g by setting - = slope . Then, solving for g gives us 9 = Slope (11) Hooke's law Hooke's law relates the force that a spring exerts on a body to the stretch of the spring. Robert Hooke (1635 - 1703), a contemporary of Isaac Newton, postulated a relationship between stress (the force applied to a spring) and strain (the stretching that results from the stress applied) Fx = -kx (12) In this equation F is the force exerted by the spring, x = |4 - Lol is the absolute value of the difference between a length of a stretched or compressed spring, L and its equilibrium length Date Modified 03/29/22 Page 3 of 10 1954 words X Text Predictions: On x Accessibility: Investigate " Focus + 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap Find Normal Title Paste Replace BI Uvab X X A LA No Spacing Heading : Heading 2 Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor Lo- The coefficient of proportionality, k, is called the spring constant, which is unique for each spring and describes how easy or hard to stretch or to compress the spring. The minus sign in the equation (12) reflects the fact that the restoring force tries to bring the spring back to its equilibrium position: compressed spring "pushes back" and stretched spring "pulls back" The equation, describing simple harmonic motion of a mass attached to an ideal spring, and the relationship between the period of oscillations and the mass attached to a light (of negligible mass) spring can be derived using Newton's Second Law. If the restoring force of the spring is the net force acting on a mass attached, then Fnet = -kx = ma (13) Remembering that the acceleration is the second derivative of position of a particle, (13) can be re-written as m-- + kx(t) = 0 (14) Equation (14) is an example of a differential equation. One solution of equation (14) is x(t) = Acos(wt). Taking the second derivative and plugging in expressions for x(t) and its second derivative into equation (14), one can see that this sets strict conditions on co. W = (15) Remembering that w = -, the period of oscillation can be expressed: T = 271 (16) OneDrive . . . X Screenshot saved The screenshot was added to your OneDrive. Page 3 of 10 1954 words X Text Predictions: On fix Accessibility: Investigate Focus 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap Find Normal Title Paste BIUvab x x ALA= = = [ No Spacing Heading : Heading 2 Replace Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor Procedures: Part I: Simple Harmonic Motion - Pendulum 1. The apparatus consists of a string, about 100 cm long, attached to a horizontal crossbar (pendulum clamp). At the lower end of the string, there is a metal sphere attached to it. 2. Measure the diameter of the sphere using a digital caliper and record it on Data Table 1. 3. Ask a lab assistant to demonstrate how to change the length of the pendulum. Change the length of the string by loosening the fastener on the pendulum clamp and adjusting the string length until it is approximately 15 cm long. 4. Measure and record the length of the pendulum in the data table. To accurately get this value, measure from the pivot point on the pendulum clamp to the top of the sphere (String Length) then add the radius of the sphere to get the total pendulum length. Record these values on Data Table 1 5. Adjust the height of the photogate as needed. The photogate should always intersect at the midpoint of the sphere. . Open the Pasco program for S1b. 7. Pull back the sphere to less than 5, release it and press the record button. Collect at least 20 seconds of data, then record the period of oscillation on Data Table 1. . Increase the height by loosening the fastener and lowering the sphere approximately 5 cm then tightening the fastener once more, and measure the new length. 9. Repeat steps 7and 8 until you record 10 trials. Part II: Hooke's Law This part of the experiment is performed to obtain the spring constant for the given spring. The horizontal cross bar should be placed approximately 60 centimeters above the table top. Adjust its height if necessary. 1. Obtain the mass of the spring (the spring by itself) and record it on the Data Sheet. 2. Place the spring back on the cross bar. Next, obtain and record the applied mass (including the mass of the mass hanger) for each of the ten trials. Start by adding a 50 gram mass to the mass hanger (trial #1), and then increase the applied mass by 50 grams for each trial. When recording these values, make sure to keep the masses in sequence (each mass is slightly different). DO NOT simply add the value written on the side of the masses to the previous total. This will give inaccurate results and require you to redo Part II. Instead, use a balance to find the mass of the hanger and the accumulated masses each time a mass is added. OneDrive . . . X . Position the meter stick next to the spring, with the 0 centimeter mark up in the air and the 100 centimeter mark on the table. Place the mass hanger and the 1: mass on the spring and record the position of the pointer (attached to the spring) relative to the meter Screenshot saved stick. Obtain the position for each of the ten trials. Use the masses that were measured before, The screenshot was added to your keeping them in the same sequence as when they were measured. OneDrive. Date Modified 03/29/22 Page 3 of 10 1954 words X Text Predictions: On x Accessibility: Investigate Focus 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap EVENEVEENT Find Normal Paste Title Replace BIUvab x x ALA = = [ No Spacing Heading : Heading 2 Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor Analysis Part I: Simple Harmonic Motion - Pendulum 1. Create a graph of the period (T2) as a function of the total length of the pendulum L- Statistically analyze the graph in order to obtain the linear regression slope for the best fit line. Be sure to remember that all calculations should be done in the standard MKS units. 3. Using the slope of the graph and equation (11), calculate the acceleration of gravity. 4. Find the percent error of your calculated acceleration "g" using 9.792 m's? as the theoretical value. Part II: Hooke's Law 1. Using the data collected from Part II of this experiment, construct a graph (using MS Excel). This graph should have the applied mass on the y-axis and the corresponding position on the x-axis. .. Statistically analyze the graph in order to obtain the linear regression slope for the best fit line. Be sure to remember that all calculations should be done in the standard MKS units. Also, it is important to note that the linear regression slope is used to calculate the spring constant; as can be seen on the attached Data Sheet - Table 2. . To obtain the spring constant (k) from the linear regression slope, multiply the slope by the experimental value for gravity you determined in Part I. OneDrive ... X Screenshot saved The screenshot was added to your OneDrive. 7 nato Mardifind 02/20/7? Page 3 of 10 1954 words X Text Predictions: On fix Accessibility: Investigate Focus 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap Find Normal Paste Title No Spacing Heading : Heading 2 Replace BIUvab x x' ALA = = [ Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor Data sheet Slb: Pendulum Harmonic Motion and Hooke's Law Data Table 1: Simple Harmonic Motion - Pendulum Diameter of Sphere (m) = 0.02533 Radius of Sphere (m) = Trial String Length (Lo) Total Pendulum Length | Period of oscillation (T) (m) (Lt = Lotr) (S ( $ 2 ) (m) 1 0.099 0.80091 2 0.126 0.87755 3 0.152 0.94059 4 0.180 0.99357 5 0.21 1.04136 6 0.238 1.09009 7 0.263 1.14575 8 0.287 1.19557 9 0.325 1.25083 10 0.350 1.29252 Slope of graph T' vs L = OneDrive ... X Gravitational Acceleration "g" = % Error = _ Screenshot saved The screenshot was added to your OneDrive. Date Modified 03/29/22 Page 3 of 10 1954 words X Text Predictions: On x Accessibility: Investigate Focus 68%W AutoSave Off S1b-Pendulum and Hookes Law 3-29-22 (1) . Saved to this PC . Search (Alt+Q) shemar tulloch ST X File Home Insert Draw Design Layout References Mailings Review View Help Comments Share k Times New Roman ~ 16 ~ A" A Aa Ap Find Normal Paste Title BIUvab x x A LAB= VL BY No Spacing Heading : Heading 2 Replace Dictate Editor Select v Undo Clipboard Font Paragraph Styles Editing Voice Editor Data sheet Slb: Pendulum Harmonic Motion and Hooke's Law Data Table 2: Hooke's Law Mass of the spring (m.): 0.008708 Applied Mass (ma) Position of Mass (x) (Kg) (m) 0.050815 0.347 0.100806 0.36 0.150765 0.381 0.200732 0.401 0.250721 0.413 0.300656 0.433 0.35104 0.457 0.40100 0.477 0.45100 0.492 0.50100 0.505 Linear regression: Slope of ma vs x = Spring constant k = OneDrive . . . X Screenshot saved The screenshot was added to your OneDrive. 10 Date Modified 03/29/22 Page 3 of 10 1954 words X Text Predictions: On x Accessibility: Investigate Focus + 68%