Chapter 14 Oscillations - University of Massachusetts Lowellfaculty.uml.edu/chandrika_narayan/Teaching/documents/Chap14Knight.… · Chapter 14 Oscillations ... Example 14.5 Using

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Chapter 14 Oscillations

Chapter Goal: To understand systems that oscillate with simple harmonic motion.

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Oscillatory Motion

Objects that undergo a repetitive motion back and forth around an equilibrium position are called oscillators.

The time to complete one full cycle, or one oscillation, is called the period T.

The number of cycles per second is called the frequency f, measured in Hz:

1 Hz = 1 cycle per second = 1 s−1

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Example 14.1 Frequency and Period of a Loudspeaker Cone

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Simple Harmonic Motion

A particular kind of oscillatory motion is simple harmonic motion.

In figure (a) an air-track glider is attached to a spring.

Figure (b) shows the glider’s position measured 20 times every second.

The object’s maximum displacement from equilibrium is called the amplitude A of the motion.

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Figure (a) shows position versus time for an object undergoing simple harmonic motion.

Figure (b) shows the velocity versus time graph for the same object.

The velocity is zero at the times when x = ± A; these are the turning points of the motion.

The maximum speed vmax is reached at the times when x = 0.

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If the object is released from rest at time t = 0, we can model the motion with the cosine function:

Cosine is a sinusoidal function.

ω is called the angular frequency, defined as ω = 2π/T.

The units of ω are rad/s.

ω = 2πf.

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The maximum speed is vmax = Αω

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Example 14.2 A System in Simple Harmonic Motion

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Example 14.3 Finding the Time

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Simple Harmonic Motion and Circular Motion

Figure (a) shows a “shadow movie” of a ball made by projecting a light past the ball and onto a screen.

As the ball moves in uniform circular motion, the shadow moves with simple harmonic motion.

The block on a spring in figure (b) moves with the same motion.

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The Phase Constant

What if an object in SHM is not initially at rest at x = A when t = 0?

Then we may still use the cosine function, but with a phase constant measured in radians.

In this case, the two primary kinematic equations of SHM are:

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Oscillations described by the phase constants φ0 = π/3 rad, −π/3 rad, and π rad.

The Phase Constant


Example 14.4 Using the Initial Conditions

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Energy in Simple Harmonic Motion

An object of mass m on a frictionless horizontal surface is attached to one end of a spring of spring constant k.

The other end of the spring is attached to a fixed wall.

As the object oscillates, the energy is transformed between kinetic energy and potential energy, but the mechanical energy E = K + U doesn’t change.

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Energy in Simple Harmonic Motion

Energy is conserved in Simple Harmonic Motion:

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Frequency of Simple Harmonic Motion

In SHM, when K is maximum, U = 0, and when U is maximum, K = 0.

K + U is constant, so Kmax = Umax:

Earlier, using kinematics, we found that:

So:

So:

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Example 14.5 Using Conservation of Energy

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Simple Harmonic Motion Motion Diagram

The top set of dots is a motion diagram for SHM going to the right.

The bottom set of dots is a motion diagram for SHM going to the left.

At x = 0, the object’s speed is as large as possible, but it is not changing; hence acceleration is zero at x = 0.

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Acceleration in Simple Harmonic Motion

Acceleration is the time-derivative of the velocity:

In SHM, the acceleration is proportional to the negative of the displacement.

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Dynamics of Simple Harmonic Motion

Consider a mass m oscillating on a horizontal spring with no friction. The spring force is:

Since the spring force is the net force, Newton’s second law gives:

Since ax = −ω2x, the angular frequency must be .

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Vertical Oscillations

Motion for a mass hanging from a spring is the same as for horizontal SHM, but the equilibrium position is affected.

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Example 14.7 Bungee Oscillations

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The Simple Pendulum

Consider a mass m attached to a string of length L which is free to swing back and forth.

If it is displaced from its lowest position by an angle θ, Newton’s second law for the tangential component of gravity, parallel to the motion, is:

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The Simple Pendulum

If we restrict the pendulum’s oscillations to small angles (< 10°), then we may use the small angle approximation sin θ ≈ θ, where θ is measured in radians.

and the angular frequency of the motion is found to be:

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Example 14.8 The Maximum Angle of a Pendulum

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Example 14.8 The Maximum Angle of a Pendulum

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Tactics: Identifying and Analyzing Simple Harmonic Motion

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The Physical Pendulum

Any solid object that swings back and forth under the influence of gravity can be modeled as a physical pendulum.

The gravitational torque for small angles (θ < 10°) is:

Plugging this into Newton’s second law for rotational motion, τ = Iα, we find the equation for SHM, with:

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Example 14.10 A Swinging Leg as a Pendulum

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Example 14.10 A Swinging Leg as a Pendulum

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Damped Oscillations

An oscillation that runs down and stops is called a damped oscillation.

One possible reason for dissipation of energy is the drag force due to air resistance.

The forces involved in dissipation are complex, but a simple linear drag model is:

The shock absorbers in cars and trucks are heavily damped springs. The vehicle’s vertical motion, after hitting a rock or a pothole, is a damped oscillation.

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Damped Oscillations

When a mass on a spring experiences the force of the spring as given by Hooke’s Law, as well as a linear drag force of magnitude |D| = bv, the solution is:

where the angular frequency is given by:

Here is the angular frequency of the undamped oscillator (b = 0).

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Damped Oscillations

Position-versus-time graph for a damped oscillator.

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Damped Oscillations

A damped oscillator has position x = xmaxcos(ωt + φ0), where:

This slowly changing function xmax provides a border to the rapid oscillations, and is called the envelope.

The figure shows several oscillation envelopes, corresponding to different values of the damping constant b.

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Mathematical Aside: Exponential Decay

Exponential decay occurs in a vast number of physical systems of importance in science and engineering.

Mechanical vibrations, electric circuits, and nuclear radioactivity all exhibit exponential decay.

The graph shows the function:

where • e = 2.71828… is

Euler’s number. • exp is the

exponential function. • v0 is called the decay constant.

u = Αe −ν/ν0 = Α exp(−ν/ν0)


Energy in Damped Systems

Because of the drag force, the mechanical energy of a damped system is no longer conserved.

At any particular time we can compute the mechanical energy from:

Where the decay constant of this function is called the time constant τ, defined as:

The oscillator’s mechanical energy decays exponentially with time constant τ.


Driven Oscillations and Resonance

Consider an oscillating system that, when left to itself, oscillates at a natural frequency f0.

Suppose that this system is subjected to a periodic external force of driving frequency fext.

The amplitude of oscillations is generally not very high if fext differs much from f0.

As fext gets closer and closer to f0, the amplitude of the oscillation rises dramatically.

A singer or musical instrument can shatter a crystal goblet by matching the goblet’s natural oscillation frequency.



The response curve shows the amplitude of a driven oscillator at frequencies near its natural frequency of 2.0 Hz.



The figure shows the same oscillator with three different values of the damping constant.

The resonance amplitude becomes higher and narrower as the damping constant decreases.


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Chapter 14 Oscillations - University of Massachusetts Lowellfaculty.uml.edu/chandrika_narayan/Teaching/documents/Chap14Knight.… · Chapter 14 Oscillations ... Example 14.5 Using