Lecture 17

Oscillations

Today’s Topics:

• Periodic motion (Simple Harmonic Motion)• Springs and pendulums• Energy• Damped and driven motion

Restoring Forces

No restoring force

Restoring force returns the ball to equilibrium

How do we describe these oscillations about equilibrium?

tAxt

Ax

wwqq

cos

cos

==

=so ,But

Simple Harmonic Motion

period T: the time required to complete one cycle

frequency f: the number of cycles per second (measured in Hz)

Tf 1=

Tf ppw 22 ==

amplitude A: the maximum displacement

tAx wcos=

ExampleThe position of a simple harmonic oscillator is given by ( )ttx 3)50.0()( p cos m =where t is in seconds. What is the period of the oscillator?

rad/s

m

3

5.0pw =

=A

tAx wcos=

Velocity and Acceleration

! tAaa

x ww cosmax

2-=

Where is vmax?

Where is amax?

Springs

xkFx -=

HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING

The restoring force on an ideal spring is

How do we determine ω?

xmakxF =-=å

tAx wcos= tAax ww cos2-=

2wmAkA -=-

mk

=w

The frequency is determined by the physical properties of the system

To measure the mass of an astronaut on the space station they employ a device that consists of a spring-mounted chair in which the astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured period is 2.41 s. Find the mass of theastronaut.

totalmk

=w

2total wkm =

( )2astrochair 2 Tkmmp

=+

( )( )( ) kg 77.2kg 0.12

4s 41.2mN606

2

2

2

chair2astro

=-=

-=

p

pm

Tkm

Springs and EnergyDEFINITION OF ELASTIC POTENTIAL ENERGY

The elastic potential energy is the energy that a springhas by virtue of being stretched or compressed. For anideal spring, the elastic potential energy is

221

elasticPE xkD=SI Unit of Elastic Potential Energy: joule (J)

22

21

21 xkmvE D+=

TotalMechanicalEnergy

As a function of time,

The total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.

Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

The Pendulum

The restoring force of the pendulum is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

Frestoring = −mgsinθ = ma

However, for small angles, sin θ and θ are approximately equal (small angle approximation)

Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. We find that the period of a pendulum depends only on the length of the string:

Damped Harmonic Motion

1) simple harmonic motion

2&3) underdamped

4) critically damped

5) overdamped

Driven Harmonic Motion and Resonance

When a force is applied to an oscillating system at all times,the result is driven harmonic motion.

Here, the driving force has the same frequency as the spring system and always points in the direction of the object’s velocity.

Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.