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1 2014 John Wiey & Sons, Inc. A rights reserved.
2 15-2 Energy in Simpe Harmonic Motion 15-2 Energy in Simpe Harmonic Motion Write the functions for kinetic and potentia energy: Eq. (15-18)18 Their sum is defined by: Eq. (15-20)18 Eq. (15-21)18 Figure John Wiey & Sons, Inc. A rights reserved.
3 15-2 Energy in Simpe Harmonic Motion 15-2 Energy in Simpe Harmonic Motion Figure 15-7 Answer: (a) 5 J (b) 2 J (c) 5 J 2014 John Wiey & Sons, Inc. A rights reserved.
4 15-3 An Anguar Simpe Harmonic Osciator 15-3 An Anguar Simpe Harmonic Osciator Learning Objectives Describe the motion of an anguar simpe harmonic osciator For an anguar simpe harmonic osciator, appy the reationship between the torque τ and the anguar dispacement θ (from equiibrium) For an anguar simpe harmonic osciator, appy the reationship between the period T (or frequency f), the rotationa inertia I, and the torsion constant κ For an anguar simpe harmonic osciator at any instant, appy the reationship between the anguar acceeration α, the anguar frequency ω, and the anguar dispacement θ John Wiey & Sons, Inc. A rights reserved.
5 15-3 An Anguar Simpe Harmonic Osciator 15-3 An Anguar Simpe Harmonic Osciator A torsion penduum: easticity from a twisting wire Moves in anguar simpe harmonic motion Eq. (15-22)18 κ is caed the torsion constant Anguar form of Hooke's aw Repace inear variabes with their anguar anaogs and we find: Eq. (15-23) John Wiey & Sons, Inc. A rights reserved. Figure 15-9
6 15-4 Penduums, Circuar Motion 15-4 Penduums, Circuar Motion Learning Objectives Describe the motion of an osciating simpe penduum Draw a free-body diagram Distinguish between a simpe and physica penduum, and reate their variabes Find anguar frequency from torque and anguar dispacement or acceeration and dispacement John Wiey & Sons, Inc. A rights reserved Distinguish anguar frequency from dθ/dt Determine phase and ampitude Describe how free-fa acceeration can be measured with a penduum For a physica penduum, find the center of the osciation Reate SHM to uniform circuar motion.
7 15-4 Penduums, Circuar Motion 15-4 Penduums, Circuar Motion A simpe penduum: a bob of mass m suspended from an unstretchabe, massess string Bob fees a restoring torque: Eq. (15-24)18 Reating this to moment of inertia: Eq. (15-26)18 Anguar acceeration proportiona to position but opposite in sign Figure John Wiey & Sons, Inc. A rights reserved.
8 2014 John Wiey & Sons, Inc. A rights reserved.
9 2014 John Wiey & Sons, Inc. A rights reserved.
10 15-4 Penduums, Circuar Motion 15-4 Penduums, Circuar Motion Anguar ampitude θ m of the motion must be sma The anguar frequency is: The period is (for simpe penduum, I = ml 2 ): Eq. (15-28)18 A physica penduum has a compicated mass distribution 2014 John Wiey & Sons, Inc. A rights reserved. Figure 15-12
11 15-4 Penduums, Circuar Motion An anaysis is the same except rather than ength L we have distance h to the com, and I wi be particuar to the mass distribution The period is: Eq. (15-29) A physica penduum wi not show SHM if pivoted about its com The center of osciation of a physica penduum is the ength L 0 of a simpe penduum with the same period 2014 John Wiey & Sons, Inc. A rights reserved.
12 2014 John Wiey & Sons, Inc. A rights reserved.
13 2014 John Wiey & Sons, Inc. A rights reserved.
14 2014 John Wiey & Sons, Inc. A rights reserved.
15 15-4 Penduums, Circuar Motion 15-4 Penduums, Circuar Motion A physica penduum can be used to determine free-fa acceeration g Assuming the penduum is a uniform rod of ength L: Then sove Eq for g: Eq. (15-30) Eq. (15-31) Answer: A the same: mass does not affect the period of a penduum 2014 John Wiey & Sons, Inc. A rights reserved.
16 15-4 Penduums, Circuar Motion 15-4 Penduums, Circuar Motion Simpe harmonic motion is circuar motion viewed edge-on Figure shows a reference partice moving in uniform circuar motion Its anguar position at any time is ωt + φ 2014 John Wiey & Sons, Inc. A rights reserved. Figure 15-15
17
18 15-4 Penduums, Circuar Motion 15-4 Penduums, Circuar Motion Projecting its position onto x: Eq. (15-36) Simiary with veocity and acceeration: Eq. (15-37) Eq. (15-38) We indeed find this projection is simpe harmonic motion 2014 John Wiey & Sons, Inc. A rights reserved.
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