A. Introduction1. Oscillations: motions that repeat themselves

a) Swinging chandeliers, boats bobbing at anchor, oscillating guitar strings, pistons in car engines

2. Understanding periodic motion essential for later study of waves, sound, alternating electric currents, and light

3. An object in periodic motion experiences restoring forces or torques that bring it back toward an equilibrium position

4. Those same forces cause the object to “overshoot” the equilibrium position

XII. Periodic Motion

1. Definitions

a) Frequency (f) = number of oscillations that are completed each second

[f] = hertz = Hz = 1 oscillation per sec = 1 s–1

b) Period = time for one complete oscillation (or cycle):

T = 1/f (XII.B.1)

XII.B. Simple Harmonic Motion (SHM)

2. Displacement x(t):

x(t) = xmcos(t + ), where (XII.B.2)

xm = Amplitude of the motion

(t + ) = Phase of the motion = Phase constant (or phase angle)) : depends on the

initial displacement and velocity = Angular frequency = 2T = 2f (rad/s) (XII.B.3)

3. Simple harmonic motion = periodic motion is a sinusoidal function of time (represented by sine or cosine function)

XII.B. Simple Harmonic Motion (SHM)

4. velocity of a particle moving with SHM:

5. The acceleration for SHM:

)sin()cos()(

)( txtxdt

d

dt

tdxtv mm

2 2( )( ) sin( ) (cos( ) )mm

dv t da t x t x t

dt

dx

t t

XII.B. Simple Harmonic Motion (SHM)

(XII.B.4)

(XII.B.5)

1. From Newton’s 2nd Law:

F = ma = –m2x (XII.C.1)

2. This result (a restoring force that is proportional to the displacement but opposite in sign) is the same as Hooke’s Law for a spring:

F = –kx, where k = m2

(XII.C.2)

XII.C. Force Law for SHM

m

k 2 / 2

mT

k (XII.C.3) (XII.C.4)

1. Elastic potential energy U = 1/2kx2 = 1/2kxm

2cos2(t + )(XII.D.1)

2. Kinetic energy K = 1/2mv2 = 1/2kxm

2sin2(t + )(XII.D.2)

3. Total mechanical energy = E = U + K E = 1/2kxm

2cos2(t + ) + 1/2 kxm2sin2(t + );

= 1/2kxm2 (XII.D.3)

XII.D. Energy in SHM

1. A simple pendulum consists of a particle of mass m (bob) suspended from one end of an unstretchable, massless string of length L that is fixed at the other enda) Consider the Forces acting on the bob:

F = –mgsin = mg(s/L); with

sin = s/LXII.E.1

b) If is small ( 150 or so) then sin :

F ~ –mg = –mgs/L. XII.E.2

XII.E. Pendula

W = mg

L

s

1. A simple pendulumsimple pendulum consists of a particle of mass m (bob) suspended from one end of an unstretchable, massless string of length L that is fixed at the other end

c) This equation is the angular equivalent of the condition for SHM (a = –2 x), so:

= (mg/L/ m)1/2 = (g/L)1/2 and (XII.E.3)

T = 2(L/g)1/2 (XII.E.4)

XII.E. Pendula

Example Problem #12A pendulum bob swings a total distance of 4.0 cm from end to end and reaches a speed of 10.0 cm/s at the midpoint. Find the period of oscillation.

xm = 0.02 m; vm = 0.10 m/s( ) sin( )mv t x t 1(0.10 ) (0.02 ).m mv ms x m

1(0.10 ) / (0.02 ) 5.0 / .ms m rad s

2 / 2 / (5.0 / ) 1.3 .T rad s s