3. Theory 3.1 Simple harmonic oscillation Many systems in nature are well approx

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3. Theory 3.1 Simple harmonic oscillation Many systems in nature are well approximated by the idealzation of simple harmonic escil tion. The word "simple" here refers to the fact that there is only one mode of osclatory motiorn rather than a combination of multiple osclations or oscilators The harmonic aspect de- scribes a restoring force that is proportional to the displacement, and the mais undengoes end less periodic motion at a single frequency. We first look at an oscillating spring with a mass m at its end, as shown in Fipure 1. Hookes aw says that the elaitic force acting on the mass is PHYS2125 Phyics Laboratory The University of Texas at Dalas where k is the spring constant, and is the diplacement of the spring how farnis stretched The negative sign physically means IhM1 that the restoring force acts always in the opposite direction of the displacement. Considering the motion in 1D and bringing in Newton's second law, we can write Because a moving object's displacement is atunction of time, x·x(t). thie above egation tells that the force and acceleration are also a funtion of timeF-Fr)anda Thus the kinematic equations we have learned for a comtant acceleration do not uapply here In fact the simple harmonic motion Mis a sinusoidal form rt)Asin(st Here A is the amplitude of oscilation and ·, the angular frequency given by From this equation of motion, one can derive the velocity and acceleration as v(r)d costut), and a(t)-A sint) The equation of motion can be generally described by X Asin(art +), with a "phase" as- sociated with the intaloondoon at r=0. but we do not realy need to consider ththsleb There are two types of mechanicall energy involved in an oscillating spring the elasti poten tial energy U(t) and kinetic enengyK)given by The elastic potential energy and the kinetic energy convert to each other periodicaly during the motion, while the total mechanical energy is conserved. Of course, real world oncillators ane not perfect and do not continue forever. External forces like friction always eventually bring them to a stop. This will be particularly true for the spring osoilator in this lab, where we wi energy loss over t 3.2 Simple pendulum Libe the simple harmonic oscilator, a simple pendulum is an ideal- zation, and it consists of a point mass im hanging from a string of length L and negigible mass|see Figure 21. The path of the mass is an arc od circle, we can use the distance x along the arc as the coord- nate, and it is related to the angle@ntadiam byx-#xL. The mass is driven by the tangential component of gravitational force, obernt Newton's 2 law, -rig sin F-ma-mgsin

Explanation / Answer

It is J/m^2 because a J is N*m and N*m/m^2 one m on top and bottom cancel to give N/m

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