Chapter 14 Oscillations and Resonance

When is the speed of the mass maximum?

DEMO In class we will observe the shadow of a point on a disk as the disk rotates.

When is the speed of the shadow maximum?

When is the speed of the shadow zero?

If the two systems require the same time for one cycle of the motion then we can say that the angular velocity of the point on the disk is equal to the angular frequency of the mass on the spring. A system completes one cycle as it moves from some starting location to some other locations and then back to that location with the same value of velocity.

page 14-3 The Sine Wave

What questions do you have about the thought experiment with the truck and billboard?

When would the shadow on the billboard have the greatest speed?

page 14-4 The period, T, is the time required for one cycle (one oscillation). T = 2π/ω

Frequency, f, is the number of complete cycles (oscillations) that occur in one second. f = 1/T

page 14-6 Phase of an Oscillation

The equations that describe the position of an object that is oscillating depend on the position of the object at time = 0 seconds. If the position is not zero then a phase angle will need to be inserted into the argument of the sine function. e.g. Y = sin(ωt + φ) φ is the phase angle. This assumes that the maximum value for Y is 1 and the minimum value is -1.

*Suppose that an object has its maximum Y value in its cycle when t = 0 seconds. What is the phase angle?

page 14-7 Mass on a Spring; Analytic Solution

Hooke’s Law F = -kX

*How many forces act on a mass that is hanging on a vertical spring?

page 14-9 To practice your calculus skills you could verify that equation 12 is a solution of equation 11 when ω= square root of (k/m) .

Since T = 2π/ω we can substitute and find T = 2π times the square root of (m/k)

Work exercise 6.

page 14-10 Amplitude is the distance the object moves from the equilibrium position to one side of the motion.

Y = A sin(ωt + φ) What is the maximum value for Y? What is the minimum value for Y?

page 14-11 Conservation of Energy

Energy is conserved for spring motion when we ignore friction. You can skip this section.

page 14-12 The Harmonic Oscillator

*Is the spring force a linear restoring force?

When the magnitude of the acceleration of the system is proportional to the displacement and there is a restoring force, then the motion is called Simple Harmonic Motion (SHM). Does the spring system qualify as SHM?

page 14-12 We will skip the Torsion Pendulum

page 14-15 The Simple Pendulum

This motion is not SHM. But, for small angles (i.e. less than 10 degrees amplitude) the motion is approximately SHM.

page 14-16 gives the result for small angles T = 2π times the square root of (length/g)

Calculate the period for the case of a pendulum 1 meter long.

Calculate the period for the case of a pendulum 9.8 meters long.

page 14-17 We will skip the conical pendulum.

page 14-18 We will skip the physical pendulum

page 14-19 We will skip Non-Linear Restoring Forces

page 14-20 Molecular Forces

Read this material.

*When can the molecular forces be approximated by a spring force?

page 14-21 Damped Harmonic Motion

When the total energy of an oscillation decreases it is said to be damped.

What happens to the amplitude of a damped harmonic oscillator as time advances?

page 14-24 Resonance

*TRUE or FALSE It is possible for a small force to create a large amplitude for harmonic motion.

We will watch a video of the Tacoma Narrows Bridge.

The frequency of a system that is oscillating on its own is called the natural frequency.

How could you perform positive work on an oscillating system at a frequency equal to the natural frequency?

What would happen to the amplitude of the oscillation if you did the above?

DEMO Resonance for the case of a mass hanging on a vertical spring.

Where is resonance important in electronic circuits?

page 14-26 Resonance Phenomena

I will have springs and masses so in class you can try the experiment discussed in the first column.

We will discuss the concepts, not the equations of this section.

Resonance exists if the frequency of the driving force is equal to the natural frequency of the system. Name some situations in which resonance is desired and some in which it is destructive.

page 14-27 Transients

We will skip this section.

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