CHAPTER15 Oscillations Ch 15 2 Simple Harmonic Motion

CHAPTER-15 Oscillations

Ch 15 -2 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)=xmcos( t+ ) = 2 f = 2 /T

Ch 15 -2 Simple Harmonic Motion

Ch 15 -2 Simple Harmonic Motion The Velocity of SHM V(t)=dx/dt= d/dt[xmcos( t+ )] v(t)=dx/dt=- xm sin( t+ ) v(t)= =-vmsin( t+ ) vm is maximum value of velocity and is called velocity amplitude The Acceleration of SHM a(t)=dv/dt= d/dt[- xm sin( t+ )] a(t)=- 2 xm cos( t+ )=- 2 x In SHM a(t)=- 2 x a(t) -x

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)

Ch 15 -4 Energy in Simple Harmonic Motion v Mechanical Energy E of a Simple Harmonic Oscillator: v E = K(t) +U(t), where K(t) and U(t) are kinetic and potential energies of the oscillator given by: v K(t)=mv 2/2=[m 2 x 2 m sin 2( t+ )]/2 v =[kx 2 m sin 2( t+ )]/2 v U(t)=kx 2/2=[kx 2 m cos 2( t+ )]/2 v E=K(t)+U(t) = kx 2 m /2

Ch 15 -5 An Angular Simple Harmonic Oscillator v Torsion Pendulum ü Disk of the pendulum oscillates in a horizontal plane with a restoring torque =- ü Then equation T=2 (m/k) ü modifies to T=2 (I/ ) , where I is moment of inertia and is torsion constant

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 = -LFgsin =-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=m. L 2; T=2 (L/g)

Ch 15 -6 Pendulums v The Physical Pendulum v For a physical pendulum , the period T=2 / = 2 (I/mgh) v where h is distance of center of mass from pivot point. v For a meterstick pivoted at one end I=ML 2/3 and h=L/2 v T=2 (2 L/3 g)