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Harmonic Motion. Chapter 13. Simple Harmonic Motion. A single sequence of moves that constitutes the repeated unit in a periodic motion is called a cycle The time it takes for a system to complete a cycle is a period (T). Simple Harmonic Motion.

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Harmonic motion

Harmonic Motion

Chapter 13


Simple harmonic motion
Simple Harmonic Motion

  • A single sequence of moves that constitutes the repeated unit in a periodic motion is called a cycle

  • The time it takes for a system to complete a cycle isa period (T)


Simple harmonic motion1
Simple Harmonic Motion

  • The period is the number of units of time per cycle; the reciprocal of that—the number of cycles per unit of time—is known as the frequency(f).

  • The SI unit of frequency is the hertz (Hz), where 1 Hz = 1 cycle/s = 1 s-1

  • Amplitude (A) is the maximum displacement of an object in SHM


Simple harmonic motion2
Simple Harmonic Motion

  • One complete orbit (one cycle) object sweeps through 2prad

  • f - number of cycles per second

  • Number of radians it movesthrough per second is 2pf - that's angular speed (w)

  • w - angular frequency

  • Sinusoidal motion (harmonic) with a single frequency - known as simple harmonic motion (SHM)



Velocity in shm
Velocity in SHM

vmax = Aw


Acceleration in shm
Acceleration in SHM

The acceleration of a simple harmonic oscillator is proportional to its displacement


Example 1
Example 1

  • A spot of light on the screen of a computer is oscillating to and fro along a horizontal straight line in SHM with a frequency of 1.5 Hz. The total length of the line traversed is 20 cm, and the spot begins the process at the far right. Determine

  • (a) its angular frequency,

  • (b) its period,

  • (c) the magnitude of its maximum velocity, and

  • (d) the magnitude of its maximum acceleration,

  • (e) Write an expression for x and find the location of the spot at t = 0.40 s.


Problem
Problem

  • A point at the end of a spoon whose handle is clenched between someone’s teeth vibrates in SHM at 50Hz with an amplitude of 0.50cm. Determine its acceleration at the extremes of each swing.


Equilibrium
Equilibrium

  • The state in which an elastic or oscillating system most wants to be in if undisturbed by outside forces.


Elastic restoring force
Elastic Restoring Force

  • When a system oscillates naturally it moves against a restoring force that returns it to its undisturbed equilibrium condition

  • A "lossless" single-frequency ideal vibrator is known as a simple harmonic oscillator.


An oscillating spring
An Oscillating Spring

  • If a spring with a mass attached to it is slightly distorted, it will oscillate in a way very closely resembling SHM.

  • Force exerted by an elastically stretched spring is the elastic restoring force F, = -ks.

  • Resulting acceleration ax = -(k/m)x

  • F is linear in x; a is linear in x - hallmark of SHM


Frequency and period
Frequency and Period

Simple harmonic oscillator

Shown every ¼ cycles for 2 cycles

Relationship between x, vx, t, and T


Hooke s law
Hooke’s Law

  • Beyond being elastic, many materials deform in proportion to the load they support - Hooke's Law


Hooke s law1
Hooke’s Law

  • The spring constant or elastic constant k - a measure of the stiffness of the object being deformed

  • k hasunits of N/m


Hooke s law2
Hooke’s Law

  • k hasunits of N/m


Frequency and period1
Frequency and Period

  • w0 - the natural angular frequency, the specific frequency at which a physical system oscillates all by itself once set in motion

    natural angular frequency

  • and since w0 = 2pf0

    natural linear frequency

  • Since T= 1/f0

    Period


Resonance vs damping
Resonance vs. Damping

  • If the frequency of the disturbing force equals the natural frequency of the system, the amplitude of the oscillation will increase—RESONANCE

  • If the frequency of the periodic force does NOT equal the natural frequency of the system, the amplitude of the oscillation will decrease--DAMPING


Example 2
Example 2

  • A 2.0-kg bag of candy is hung on a vertical, helical, steel spring that elongates 50.0 cm under the load, suspending the bag 1.00 m above the head of an expectant youngster. The candy is pulled down an additional 25.0 cm and released. How long will it take for the bag to return to a height of 1.00 m above the child?


The pendulum
The Pendulum

  • The period of a pendulum is independent of the mass and is determined by the square root of its length


Example 3
Example 3

  • How long should a pendulum be if it is to have a period of 1.00 s at a place on Earth where the acceleration due to gravity is 9.81 m/s2?


Problem 2
Problem 2

  • What would the length of a pendulum need to be on Jupiter in order to keep the same time as a clock on Earth? gJupiter = 25.95m/s2


When “f” is known

When “f” is NOT known

vmax = Aw


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