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

ACT: to the center of Earth

A hole is drilled through the center of Earth and emerges on the other side. You jump into the hole. What happens to you ?

a) you fall to the center and stopb) you go all the way through and

continue off into spacec) you fall to the other side of

Earth and then returnd) you won’t fall at all

You fall through the hole. When you reach the center, you keep going because of your inertia. When you reach the other side, gravity pulls you back toward the center. Gravity is a restoring force here!

Follow-up: Where is your acceleration zero?

How do we describe 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=

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T?

a) 0b) A/2c) Ad) 2Ae) 4A

The displacement is Δx = x2 – x1. Because the initial and final positions of the mass are the same (it ends up back at its original position), then the displacement is zero.

ACT: simple harmonic motion

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 for a spring?

xmakxF =-=å

tAx wcos= tAax ww cos2-=

2wmAkA -=-

mk

=w

The frequency is determined by the physical properties of the system

A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change?

a) period will increaseb) period will not changec) period will decrease

The period of simple harmonic motion depends only on the mass and the spring constant and does not depend on the acceleration due to gravity. By going to the Moon, the value of g has been reduced, but that does not affect the period of the oscillating mass–spring system.

ACT: spring on the Moon

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 the astronaut.

totalmk

=w

2total wkm =

( )2astrochair 2 Tkmmp

=+( )

( )( ) kg 77.2kg 0.124

s 41.2mN606

2

2

2

chair2astro

=-=

-=

p

pm

Tkm

Weighing when apparently weightless

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

Frestoring = −mgsinθ = ma

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

However, for small angles, sinθ 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 on the length of the string and g:

Two pendulums have the same length, but different masses attached to the string. How do their periods compare?

a) period is greater for the greater massb) period is the same for both casesc) period is greater for the smaller mass

ACT: period of a pendulum

The period of a pendulum depends on the length and the acceleration due to gravity, but it does not depend on the mass of the bob.

Follow-up: What happens if the amplitude is doubled?

gLT = 2π

The acceleration due to gravity is smaller on the Moon. The relationship between the period and g is given by:

therefore, if g gets smaller, T will increase.gL

T = 2π

a) period increasesb) period does not changec) period decreases

A swinging pendulum has period T on Earth. If the same pendulum were moved to the Moon, how does the new period compare to the old period?

ACT: pendulum on the moon

A grandfather clock has a weight at the bottom of the pendulum that can be moved up or down. If the clock is running slow, what should you do to adjust the time properly?

a) move the weight upb) move the weight downc) moving the weight will not matterd) call the repairman

ACT: grandfather clock

The period of the grandfather clock is too long, so we need to decrease the period (increase the frequency). To do this, the length must be decreased, so the adjustable weight should be moved up in order to shorten the pendulum length.

gL

T = 2π

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.