NAME ID# DATE LAB PARTNERS LAB INSTRUCTORS LAB # TITLE OBJECTIVE : : : :

Anil Rochester : : 18085026 January 26, 2011

All General Physics Wednesday lab students. Mr. Cedric Hanson 2 Simple Harmonic Motion-Mass on a Spring

a) To determine the extent to which the frequency of the mass/spring system oscillations is independent of the amplitude of the oscillations. b) To test the dependence of the period of the oscillations on mass and spring constant, as summarized by the equation: T = 2 m/k, where k is the spring constant, m is the oscillating mass, and T is the period of the oscillator.

APPARATUS SETUP

Procedure: The procedure was set up in a similar way as shown in the apparatus setup above. PART I Spring Constant 1) The weight hanger was set up and suspended from a vertically hanging spring. A reasonable range of masses for the spring was chosen, which was between 50-200gs. 2) A reading was taken of the height of the bottom of the hanger when no masses are in it. This was the zero point and was recorded as 0.725m. 3) A mass of 50g was added and the height was recorded to be 70.9 cm. This was repeated four times and recorded to be 69.2cm at 100g, 67.5cm at 150g, 65.9cm at 200g. 4) The masses were removed in the opposite order that they were placed. PART II Effect of Amplitude on Period 5) The same four masses were used to perform the following experiment. 6) The mass of the hanger and the spring were measured and recorded to be 28.4g and 7.7g respectively. 7) With a mass between 50g and 100g in the hanger, the spring was lifted slightly then released and the mass and oscillations were observed. 8) The period of oscillations for two different amplitudes were recorded using the same mass. The amplitudes were moved first gently then heavily and recorded in the data table. PART III Effect of Mass on Period 9) For each of the four masses used in Part I they were each divided by 30 to give the period of oscillation, T. 10) Three trials of each mass were taken and averaged for the best value of T. This was documented in data table. PRECAUTIONS

The clamp stand was held so that the iron could be fastened to avoid the set-up from falling apart. It was ensured that the spring was still before reading and recording the extension. Each piece of measuring equipment was inspected to guarantee that an accurate reading was obtained. The experiment was conducted repeatedly to ensure accuracy. The equipment was treated with utmost care in order to prevent damage to it or the students operating it. A reasonable range was determined and the spring did not become permanently stretched.

APPARATUS: Spiral spring Meter stick Stopwatch Electronic balance Weights and hanger Stand and clamp

ACCURACY OF ERRORS: Inaccuracies exist in the measuring of Simple Harmonic Motion and Mass on a spring The weights fell off several times. The spring was bobbing too fast and that made it hard for the oscillations to be counted. The zero point was not recorded accurately. Reaction time was not accounted for in each trial. It also varied.

DISCUSSION: According to the data collected, the periods do not differ by more than can be accounted for by experimental error, as period does not depend on the amplitude. The period of an oscillation depends on the mass m and stiffness constant k, but not the amplitude A. The larger the mass, the more inertia is in the system, which causes a slower response, which in turn causes smaller acceleration. However, the larger (k) is the greater the force and quicker the response resulting in larger accelerations. Motion that repeats itself in equal intervals of time is called harmonic motion which relates to the experiment of simple harmonic motion. When an object moves back and forth over the same path in harmonic motion it is said to be oscillating. If the amount of motion of an oscillating object (the distance the object travels) stays the same during the period of motion, it is called simple harmonic motion (SHM). The oscillating object does not lose any energy in SHM, showing that the system conserved energy and friction is assumed to be zero. In simple harmonic motion there is always a restorative force, which acts in the opposite direction of the velocity; this concept was proven during the experiment as the spring swung back and forth. The restorative force changes during oscillation and depends on the position of the object. In a spring the force is the Hookes Law, F= -kx. When y = 0 the mass is at the center or equilibrium position. The restoring force must exist for both the positive and negative y-axis. The resultant equation of motion of the mass (variation of y with time) is given starting from y = 0 at t = 0. As a spring compresses, the force (and hence acceleration) increases. As a pendulum swings, the tangential component of the force of gravity changes, so the acceleration changes. In SHM, T is the time it takes the object to return to its exact starting point and starting direction. T = 1 / f; Period and frequency are inversely related and the period of an oscillation in seconds for a mass oscillating on a spring depends on the mass of the object on the spring and the spring constant. The frequency, f, is the number of cycles an object goes through in 1 second. Frequency is measured in Hertz (Hz). 1 Hz = 1 cycle per sec. The amplitude, A, is the distance from the equilibrium (or center) point of motion to either its lowest or highest point (end points). The amplitude, therefore, is half of the total distance covered by the oscillating object and can vary in harmonic motion but is constant in SHM. The kinetic energy and the speed are at a maximum at

the equilibrium point, but the potential energy and restorative force is zero there. At the end points the potential energy is at a maximum, while the kinetic energy and speed are zero. However at the end points the restorative force and acceleration are at a maximum. Based on the experiment, the values obtained fall very close to the theoretical values as explained by the text. In procedure I, the data collected for both small and large amplitude were compared to the theoretical value, for example small vs. large amplitude in (a) 0.80 and 0.77 respectively, had differences of approximately 0.03 which can be attributed to parallax errors during the experiment. Therefore, procedure I, confirmed the statement that the period does not depend on the amplitude. For procedure II, the graph of applied forces vs. extensions in meters was plotted using data in the results section and a straight line was obtained, this graph proved that force is directly proportional to displacement. In procedure III, the Graph of Period squared vs. mass also resulted in a straight line which confirms the theoretical aspect with respect to the relationship between period and mass, this states that the period of a simple harmonic oscillator is found to depend on the stiffness of the spring and also the mass that is oscillating. Therefore, the larger the mass the more inertia is applied to the system, thereby causes a decrease in acceleration.

CONCLUSION: In concluding, the objectives of this experiment were successfully carried out, as the theories behind the experiment were proven to be valid over a wide range of observations. Within the limits of experimental error the extensions were found to be proportional to the force applied to the system and also the mass applied was directly proportional to the period squared. The experiment also proved that the frequency is completely independent of amplitude for a true harmonic oscillator.

RESULTS: Procedure #1: Weight holder = 11.0 g Weight added = 70.0 g Weight holder + weight added = 81.0 g Length of spring = 23.5 cm or 0.235m Mass of spring = 10.5 g Small amplitude: Period (1cm 12 cm) a. 0.80 b. 0.78 c. 0.79 Large Amplitude: Period (A 12cm) a. 0.77 b. 0.77 c. 0.77 Procedure #2: Mass (kg) 0.00 0.011 0.021 0.031 0.041 0.051 0.061 Procedure #3: Mass of spring = 10.5 g Mass (kg) 0.061 0.071 0.081 0.091 0.101 Period (s) 0.58 0.69 0.79 0.84 0.88 Period 2/ T2 (s)2 0.34 0.48 0.62 0.71 0.77 Force (N) 0.00 0.11 0.21 0.30 0.40 0.50 0.60 Extension (m) 0.00 1.3 10-3 2.3 10-3 5.5 10-3 9.5 10-3 1.75 10-2 2.95 10-2

N.B. THESE RESULTS WERE COLLECTED AND NOT FROM DOING THE EXPERIMENT AS I WAS ABSENT FROM THE LAB ATTENDING MY FATHER'S FUNERAL

Diagrams

Spring

Meter Stick

Clamp and Masses

Electric Balance

Chronometer

Post lab questions:

1. Using the laws of physics, (equations not required), explain why the period of an oscillator should increase as the oscillating mass increases. The period of an oscillatory system should increase, because to stretch or compress a spring work has to be done, hence potential energy is stored in a stretched or compressed spring. An increase in mass would provide greater potential energy resulting in greater kinetic energy, which should increase the oscillating mass. Now, the total mechanical energy of a spring mass system is the sum of the kinetic energy and potential energies which is considered to be conserved. 2. In the experiment, the mass of the spring was not taken into account. Describe what effect the inclusion of all or a portion of the springs mass would have on the results for T. Similarly, to question one, the mass increase would increase the period as the mass of the spring will add to the mass of the oscillating object. Hence, the resuts for T would have increased.