- Simple pendulums Energy conservation 3. Theory 3.1 Simple harmonic oscillation

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- Simple pendulums Energy conservation 3. Theory 3.1 Simple harmonic oscillation Many systems in nature are well approximated by the idealization of simple harmonic oscll tion The wond "simple" here efers to the fact that there is only one mode of oscilatory motion rather than a combination of mutole osollators or osclators. The armor.ont d scribes a restoring force thart is proportional to the dsplacement, and the mass undrgos end less periodic motion at a single frequency we first look at an oscilating terne with a mass m at its end, as shown aw says that the elastic force acting on the mass is Fe n 1 Hooke's PHYS2125 Physics Laboratory The University of Texas at Dalan where k the tering constant, andis the diselacement of the spring (how far it is stretched). The negative sign physically means that the restoreforce acts always in, the opposite deection of the displacement. Considering the motion in 1D and bringing in Newton's second law, we can write Becausemoving obrers ásplacement is ,unction or tirne, x att). the above en tels that thie force and axeleration are also a function of tme, F·F(r) and a·a(r). Th the kinematic equations have learned for constant acceleration do not apply iwa ln the simple harmonic motion has a sinusoidal form r(t)Asinut) Here A is the amplitude of oscillation and is the angular frequency given by From this equation of motion, one can derive the velocity and acceleration as The equation of motion can be grneralydescribed byr.Astfut++), with a "phase-a sociated with the iitial cond ton "0,but w do not really need toconsider leb There are two types of mechanical energy involved inoscilating spmfthe eir tial energy Ut) and kinetic energy Kt), even by p The elastic potential energy and the kinetic energy convert to each other periodically during the total mechanical energy is s conserved. Of course eal world asctosa motion, while not perfet and do net continue forever. External forces ke friction always eventually bring them to a stop. This willbe particularly true for the spring escllator in this lab, whee wew observe the energy loss over tine 3.2 Simple pendulum Like the simple harmonic oscilator, a simple pendulum is an ideal ation, and sists of a point mass m hanging trom a string of length L and negig ble mass (see Figure 2). The path of the mass is an anc of a circle. We can use the dstance x along the anc as the coord nate, andRis related to the angle@nradans byr.@xL The mass es driven bythe tangential component of gravitational force, obemg Newton's 2 a -ngsne

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