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Simple Harmonic Motion. An Introduction. Periodic Motion. Repeated motion along the same path over a fixed reproducible period of time Acrobat on a trapeze Wrecking ball Playground swing Mass attached to a spring. x. m. L. x = -A. x = 0. x = +A. Oscillators.

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simple harmonic motion

Simple Harmonic Motion

An Introduction

periodic motion
Periodic Motion

Repeated motion along the same path over a fixed reproducible period of time

  • Acrobat on a trapeze
  • Wrecking ball
  • Playground swing
  • Mass attached to a spring
oscillators

x

m

L

x = -A

x = 0

x = +A

Oscillators
  • Mechanical devices that create this motion
  • Weight attached to spring vertically
  • Weight attached to spring horizontally
  • Pendulum
if you were to plot distance vs time you would get a graph that resembled a sine or cosine function
If you were to plot distance vs time you would get a graph that resembled a sine or cosine function.

3

t(s)

2

4

6

-3

x(m)

simple harmonic motion shm
Simple Harmonic Motion (SHM)
  • Periodic motion which can be described by a sine or cosine function.
  • Springs and pendulums are common examples of Simple Harmonic Oscillators (SHOs).
equilibrium
Equilibrium
  • The midpoint of the oscillation of a simple harmonic oscillator.
  • Position of minimum potential energy and maximum kinetic energy.
u g increases as pendulum s displacement increases
Ug increases as pendulum’s displacement increases

Maximum

Displacement

Maximum

Displacement

Equilibrium

u s increases as pendulum s displacement increases
Us increases as pendulum’s displacement increases

Maximum

Displacement

Maximum

Displacement

Equilibrium

m

m

m

all oscillators obey
All oscillators obey…

Law of Conservation of Energy

amplitude a

Amplitude A

Amplitude (A)
  • How far the body is from equilibrium at its maximum displacement.
  • Higher amplitude equals more energy
period t
Period (T)
  • The length of time it takes for one cycle of periodic motion to complete itself.
frequency f
Frequency (f):
  • How fast the oscillation is occurring.
  • Frequency is inversely related to period.
  • f = 1/T
  • The units of frequency is the Hertz (Hz) where 1 Hz = 1 s-1.
slide14

x

F

Example 1:The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion?

Period: T = 0.500 s

Frequency: f = 2.00 Hz

parts of a wave

T

A

Equilibrium point

Parts of a Wave

3

t(s)

2

4

6

-3

x(m)

springs
Springs
  • A very common type of Simple Harmonic Oscillator.
  • Our springs will be ideal springs.
    • They are massless.
    • They are compressible and extensible.
  • They are restored to equilibrium according to Hooke’s Law: Fs=-kx
restoring force
Restoring force
  • The restoring force is the secret behind simple harmonic motion.
  • The force is always directed so as to push or pull the system back to its equilibrium (normal rest) position.
hooke s law
Hooke’s Law

A restoring force directly proportional to displacement is responsible for the motion of a spring.

F = -kx

where

F: restoring force

k: force constant

x: displacement from equilibrium

hooke s law1

Fs

m

mg

Hooke’s Law

The force constant of a spring can be determined by attaching a weight and seeing how far it stretches.

Fs = -kx

hooke s law2

x

m

F

m

F

x

m

Hooke’s Law

F = -kx

Equilibrium position

Spring compressed, restoring force out

Spring at equilibrium, restoring force zero

Spring stretched, restoring force in

a 4 kg mass suspended from a spring produces a displacement of 20 cm what is the spring constant

0.20m

F

m

DF

Dx

39.2 N

0.2 m

k = =

A 4-kg mass suspended from a spring produces a displacement of 20 cm. What is the spring constant?

The stretching force is the weight (W = mg) of the 4-kg mass:

F = (4 kg)(9.8 m/s2) = 39.2 N

Now, from Hooke’s law, the force constant k of the spring is:

k = 196 N/m

the mass m is now stretched a distance of 8 cm and held what is the potential energy k 196 n m

0.08 m

F

0

m

The mass m is now stretched a distance of 8 cm and held. What is the potential energy? (k = 196 N/m)

The potential energy is equal to the work done in stretching the spring:

U = 0.627 J

acceleration in shm

+a

-a

-x

+x

m

x = -A

x = 0

x = +A

Acceleration in SHM
  • Acceleration is in the direction of the restoring force. (a ispositive when x is negative, and negative when x is positive.)
  • Acceleration is a maximum at the end points and it is zero at the center of oscillation.
acceleration vs displacement

a

v

x

m

x = -A

x = 0

x = +A

Acceleration vs. Displacement

Given the spring constant, the displacement, and the mass, the acceleration can be found from:

or

Note: Acceleration is always oppositeto displacement.

slide25

a

+x

m

A 2-kg mass hangs at the end of a spring whose constant is k = 400 N/m. The mass is displaced a distance of 12 cm and released. What is the acceleration at the instant the displacement is x = +7 cm?

a = -14.0 m/s2

Note: When the displacement is +7 cm (downward), the acceleration is -14.0 m/s2 (upward) independent of motion direction.

what is the maximum acceleration for the 2 kg mass in the previous problem a 12 cm k 400 n m

+x

m

What is the maximumacceleration for the 2-kg mass in the previous problem? (A = 12 cm, k = 400 N/m)

The maximum acceleration occurs when the restoring force is a maximum; i.e., when the stretch or compression of the spring is largest.

xmax=  A

F = ma = -kx

Maximum Acceleration:

amax = ± 24.0 m/s2

period of a spring
Period of a spring
  • T = 2m/k
    • T: period (s)
    • m: mass (kg)
    • k: force constant (N/m)
slide28

m

a

v

x

x = 0

x = -0.2 m

x = +0.2 m

The frictionless system shown below has a 2-kg mass attached to a spring (k = 400 N/m). The mass is displaced a distance of 20 cm to the right and released.What is the frequency of the motion?

f = 2.25 Hz

pendulums
Pendulums
  • Springs are not the only type of oscillators.
  • The pendulum can be thought of as an oscillator.
slide30
What factor(s) determine the period of a pendulum??A) MassB) Length of StringC) GravityD) Angle of displacement
the simple pendulum

L

T

mg

The Simple Pendulum

The period of a simple pendulum is given by:

For small angles q.

slide32

L

L = 0.993 m

What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)?