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Simple Harmonic Motion

Simple Harmonic Motion - PHYSICS 215 SIMPLE HARMONIC MOTION...

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1 PHYSICS 215 SIMPLE HARMONIC MOTION Object: The object of this experiment is to predict and measure properties of systems moving in simple harmonic motion: a simple pendulum; a physical pendulum; and a linear mass and springs oscillator. Apparatus: Rotary Motion Sensor Rod and Masses 2 m stick Air Track Air Cart with Springs Triple Beam Balance Spherical mass and string Ultrasonic Motion Sensor 1.0 Introduction Periodic motion, from masses on springs to vibrations of atoms, is one of the most important kinds of physical behavior. In this lab, we will deal with Hooke’s law, where the force is proportional to the displacement, tending to restore objects to some equilibrium position. A large number of physics systems can be successfully modeled with this simple idea, including the vibrations of strings, the swinging of a pendulum, and the propagation of waves of all kinds. All of these phenomena involve periodic motion. Periodic vibrations can cause disturbances that move through a medium in the form of waves. Many kinds of waves occur in nature, such as sound waves, water waves, seismic waves, and electromagnetic waves. These very different physical phenomena are described by common terms and concepts that will be introduced in this lab. 2.0 Theory 2.1 Simple Harmonic Motion One important property of oscillator (periodic) motion is called its frequency, or number of oscillations that are completed each second. The symbol for frequency is f, and the SI unit is hertz (abbreviated Hz), where 1 hertz = 1 Hz = 1 oscillation per second = 1 s -1 (1) Related to the frequency is the period T of the motion, where period is the time for one complete oscillation (or cycle ); that is ܶ ൌ 1 ݂ (2) Any motion that repeats itself at regular intervals is called periodic motion or harmonic motion. We are interested in the displacement x of a particle from the origin given as a function of time by ݔሺݐሻ ൌ ݔ cosሺ߱ݐ ൅ ߶ሻ (3) where x m , ω , and φ are constants. This motion is called simple harmonic motion (SHM), a term that means the periodic motion is a sinusoidal function of time. Equation (3) is the sinusoidal function of a cosine function shown in Figure 1.
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2 Figure 1 A graph of equation (3) taken from Wikipedia. Figure 2 A handy reference to the quantities in equation (3) for simple harmonic motion. The quantity x m , called the amplitude of the motion, is a positive constant whose value depends on how the motion was started. The subscript m stands for maximum because the amplitude is the magnitude of the maximum displacement of the particle in either direction. The cosine function in equation (3) varies between the limits ±1; so the displacement x(t) varies between the limits ±x m . The time-varying quantity ) ( φ ω + t in equation (3) is called the phase of the motion, and the constant φ is called the phase constant (or phase angle) . The value of φ depends on the displacement and velocity of the particle at time t=0.
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