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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 isa period (T)

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 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

vmax = Aw

Acceleration in SHM

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

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

- 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

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

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

- 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

Simple harmonic oscillator

Shown every ¼ cycles for 2 cycles

Relationship between x, vx, t, and T

Hooke’s Law

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

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 Law

- k hasunits of N/m

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

- 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

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

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

- 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

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