Physics 170 - MechanicsLecture 31

Oscillations I

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Oscillations

Oscillations (whether sinusoidal or otherwise) have some common characteristics:1. They take place around an equilibrium position;2. The motion is periodic and repeats with each cycle.

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Periodic MotionPeriod: time required for one cycle of periodic motion

Frequency: number of oscillations per unit time

The frequency unit is called a hertz (Hz):

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Frequency and Period

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Example: Radio Station Frequency and Period

What is the oscillation period of an FM radio station that broadcasts at 100 MHz?

Note that 1/Hz = s

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Simple Harmonic Motion

A spring exerts a restoring force that is proportional to the displacement from equilibrium:

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Simple Harmonic Motion A mass on a spring has a displacement as a function of time that is a sine or cosine curve: Here, A is called the amplitude of the motion.

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Simple Harmonic Motion If we call the period of the motion T (this is the time to complete one full cycle) we can write the position as a function of time as:

It is then straightforward to show that the position at time t + T is the same as the position at time t (one period earlier), as we would expect.

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Connections between Uniform Circular Motion and Simple Harmonic Motion

An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:

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Uniform circular motion projected into one dimension is simple harmonic motion (SHM).

Consider a particle rotating with the angle φ increasing linearly with time:

Connections between Uniform Circular Motion and Simple Harmonic Motion

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Connections between Uniform Circular Motion and Simple Harmonic Motion

Here, the object in circular motion has an angular speed of

where T is the period of motion of the object in simple harmonic motion.

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Connections between Uniform Circular Motion and Simple Harmonic Motion

The position as a function of time:

The angular frequency:

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Connections between Uniform Circular Motion and Simple Harmonic Motion

The velocity as a function of time:

And the acceleration:

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The Period of a Mass on a Spring Since the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that .

Substituting the time dependencies of a and x gives:

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The Period of a Mass on a SpringTherefore, the period is:

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Mass+SpringSimple Harmonic Motion

In simple harmonic motion (SHM), the acceleration, and thus the net force, are both proportional to and oppositely directed from the displacement from the equilibrium position.

A = amplitudeω = angular frequencyδ = phase

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SHM Prototype Experiment Consider Fig. (a). An air-track glider attached to a spring. The glider is pulled a distance A from its rest position and released. Fig. (b) shows a graph of the motion of the glider, as measured each 1/20 of a second.

The graphs on the right show the position and velocity of the glider from the same measurements. We see that A=0.17 m and T=1.60 s. Therefore the oscillation frequency of the system is f = 0.625 Hz

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Two Oscillating Systems The diagram shows two identical masses attached to two identical springs and resting on a horizontal frictionless surface. Spring 1 is stretched to 5 cm, spring 2 is stretched to 10 cm, and the masses are released at the same time.

Which mass reaches the equilibrium position first?

Because k and m are the same, the systems have the same period, so they must return to equilibrium at the same time.

The frequency and period of SHM are independent of amplitude.

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Example: A Block on a Spring A 2.00 kg block is attached to a spring as shown.The force constant of the spring is k = 196 N/m.The block is held a distance of 5.00 cm fromequilibrium and released at t = 0.

(a) Find the angular frequency ω, the frequency f, and the period T.(b) Write an equation for x vs. time.

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Example: A System in SHMAn air-track glider is attached to a spring,

pulled 20 cm to the right, and releasedat t-=0. It makes 15 completeoscillations in 10 s.

a. What is the period of oscillation?b. What is the object’s maximum speed?c. What is its position and velocity at t=0.80 s?

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The Phase Constant But what if φ is not zero at t=0?

A phase constant φ≠0 means that the rotation starts at a different point on the circle, implying different initial conditions.

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SHM Initial Conditions

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Phases and Oscillations

Here are three examples of differing initial conditions:

1. φ0 = π/3, implying x0 = A/2 and moving to the left (v<0);

2. φ0 = −π/3, implying x0 = A/2 and moving to the right (v>0);

3. φ0 = π, implying x0 = −A and momentarily at rest (v=0).

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