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CHAPTER-15 Oscillations
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Ch 15-2 Simple Harmonic Motion Simple Harmonic Motion (Oscillatory motion) back and forth periodic motion of a particle about a point (origin of an axis). Frequency f: Number of oscillations completed each second Period of the motion T= 1/f Displacement x of the particle about the origin is a sinusoidal function of time t given by : x(t)=x m cos( t+ ) = 2 f = 2 /T
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Ch 15-2 Simple Harmonic Motion
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The Velocity of SHM V(t)=dx/dt= d/dt[x m cos( t+ )] v(t)=dx/dt=- x m sin( t+ ) v(t)= =-v m sin( t+ ) v m is maximum value of velocity and is called velocity amplitude The Acceleration of SHM a(t)=dv/dt= d/dt[- x m sin( t+ )] a(t) =- 2 x m cos( t+ )=- 2 x In SHM a(t) =- 2 x a(t) -x
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Ch 15-3 Force Law for Simple Harmonic Motion Force Required for SHM F=ma =m(- 2 x)=-(m 2 )x=-kx familiar restoring force of Hook’s law: Spring force with spring constant k= m 2 Block-spring system forms linear simple harmonic oscillator with angular frequency = (k/m) ; Oscillation Period T=2 / = 2 (m/k)
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Mechanical Energy E of a Simple Harmonic Oscillator: E = K(t) +U(t), where K(t) and U(t) are kinetic and potential energies of the oscillator given by: K(t)=mv 2 /2=[m 2 x 2 m sin 2 ( t+ )]/2 =[kx 2 m sin 2 ( t+ )]/2 U(t)=kx 2 /2=[kx 2 m cos 2 ( t+ )]/2 E=K(t)+U(t) = kx 2 m /2 Ch 15-4 Energy in Simple Harmonic Motion
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Ch 15-5 An Angular Simple Harmonic Oscillator Torsion Pendulum Disk of the pendulum oscillates in a horizontal plane with a restoring torque =- Then equation T=2 (m/k) modifies to T=2 (I/ ), where I is moment of inertia and is torsion constant
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Ch 15-6 Pendulums The Simple Pendulum consists of a particle of mass m (called bob of the pendulum) suspended from one end of an unstretchable, massless string of length L. The bob back and forth motion under a restoring torque ( =r F). Then = -LF g sin =-Lmgsin = I ; For small values of , sin = Then I = =-Lmg and = -(Lmg/I) =- 2 ; 2 = L mg/I T=2 / = 2 (I/Lmg). For a simple pendulum I=mL 2 ; T=2 (L/g)
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Ch 15-6 Pendulums The Physical Pendulum For a physical pendulum, the period T=2 / = 2 (I/mgh) where h is distance of center of mass from pivot point. For a meterstick pivoted at one end I=ML 2 /3 and h=L/2 T=2 (2L/3g)
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