WarmupA uniform beam 2.20m long with mass

m=25.0kg, is mounted by a hinge on a wall as shown. The beam is held horizontally by a wire that makes a 30° angle as shown. The beam supports a mass M = 280kg suspended from its end. Determine the components of the force FH that the hinge exerts and the components tension, FT in the supporting wire.

30°

M

Harmonic Motion

Simple Harmonic Motion

When a vibration or oscillation repeats itself, back and forth, over the same path, the motion is periodicAny vibrating system in which the restoring force is directly proportional to the negative of the displacement is said to exhibit simple harmonic motion

Such a system is often called a simple harmonic oscillator A mass oscillating on the end

of a spring is an example of a simple harmonic oscillator

The Motion of an Oscillating Object

The motion of an object undergoing simple harmonic oscillation is sinusoidal in nature

x(t) = A sin (2πt/T)

The Total Energy of a Vibrating System is Constant

KE + PE = constantIf the maximum amplitude of the motion is x0 then the energy at any point x is given by:

½ mv2 + ½ kx2 = ½ kx02

From this we can solve for velocity:│v│= √ [(x0

2 –x2)(k/m)]From Hooke’s law, F = -kx and F =ma, therefore

a = -(k/m) x

Reference CirclePoint P moves with constant velocity v0 around the circlePoint A is the projection of point P on the x axisThe motion of A back and forth about point O is second harmonic motionThe time for P to go around the circle is TThe velocity of point A is

v = -v0 sin θ

P

A

r= x0

θ

θ

Displacement x

Once around in time T

v0

v

O

Reference CircleThe period T is:

T = 2πr0/v0 = 2πx0/v0

But v0 is the maximum speed of point A Maximum speed occurs when x = 0

Therefore, v0 = x0 √(k/m)Which makes T = 2π√(m/k)And, since f = 1/T,

f = (1/2π)√(k/m)

Period and Frequency of an Oscillator

The period of a simple harmonic oscillator is given by:

T = 2π√(m/k)Therefore, since f = 1/T,

f = (1/2π)√(k/m)

Simple Pendulums

For a pendulum the restoring force is

F = -mg sin θ, But for small displacements, F = -mg sin θ ~ -mgθ

But since x = Lθ we haveF ~ (mg/L) x

Therefore the motion is essentially harmonic

Simple PendulumsBut if a pendulum is an harmonic oscillator, then k = mg/L, therefore

T = 2π√ (m/mg/L) = 2π√(L/g) and

F = 1/T = 1/(2π)√(g/L)The frequency of a pendulum does not depend on the mass of the pendulum bob! Consider the case of large and small children

on a swing—the period remains the same

In Real Life Most Harmonic Motion is Damped

Natural oscillations decrease with time due to friction and other lossesSometimes the damping is so great that the motion does not even appear to be harmonic

A = overdampedB = critically dampedC = underdamped

A shock absorber is a damped oscillator

Shock Absorbers Keep a Car From Oscillating

LinksJava simulation—springJava simulation-oscillatorSimple Harmonic Motion and Uniform Circular Motion

Do Now (10/8/13):Suppose that a pendulum has a period of 1.5 seconds. How long does it take to make a complete back-and-forth vibration? Is this 1.5 s period pendulum longer or shorter in length than a 1 s period pendulum?

Practice:Brainstorm answers to the Conceptual Questions in Chapter 13 (10 min)Complete the Multiple Choice questions in Chapter 13 before the end of class.

Do Now (10/9/13):Pass in your HW then wait quietly for instructions