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OSCILLATIONS ABOUT EQUILIBRIUM Chapter 13
Copyright © 2010 Pearson Education, Inc.
• Periodic Motion
• Simple Harmonic Motion
• Connections between Uniform Circular Motion and Simple Harmonic Motion
• The Period of a Mass on a Spring
• Energy Conservation in Oscillatory Motion
• The pendulum
Units of Chapter 13
Copyright © 2010 Pearson Education, Inc.
13-1 Periodic motion Copyright © 2010 Pearson Education, Inc.
Period: time required for one cycle of periodic motion
Frequency: number of oscillations per unit time
This unit is called the Hertz:
13-2 Simple Harmonic Motion Copyright © 2010 Pearson Education, Inc.
A spring exerts a restoring force that is proportional to the displacement from equilibrium:
13-2 Simple Harmonic Motion Copyright © 2010 Pearson Education, Inc.
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.
13-2 Simple Harmonic Motion Copyright © 2010 Pearson Education, Inc.
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:
It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect.
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
Copyright © 2010 Pearson Education, Inc.
An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
Copyright © 2010 Pearson Education, Inc.
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.
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
Copyright © 2010 Pearson Education, Inc.
The position as a function of time:
The angular frequency:
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion
Copyright © 2010 Pearson Education, Inc.
The velocity as a function of time:
And the acceleration:
Both of these are found by taking components of the circular motion quantities.
13-4 The Period of a Mass on a Spring
Copyright © 2010 Pearson Education, Inc.
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
13-4 The Period of a Mass on a Spring
Copyright © 2010 Pearson Education, Inc.
Therefore, the period is
13-5 Energy conservation in oscillatory motion Copyright © 2010 Pearson Education, Inc.
In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:
Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:
13-5 Energy conservation in oscillatory motion Copyright © 2010 Pearson Education, Inc.
As a function of time,
So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.
13-5 Energy conservation in oscillatory motion Copyright © 2010 Pearson Education, Inc.
This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.
Example Copyright © 2010 Pearson Education, Inc.
The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point?
Example continued Copyright © 2010 Pearson Education, Inc.
Example Copyright © 2010 Pearson Education, Inc.
The diaphragm of a speaker has a mass of 50.0 g and responds to a signal of 2.0 kHz by moving back and forth with an amplitude of 1.8×10−4m at that frequency.
Example continued Copyright © 2010 Pearson Education, Inc.
Example Copyright © 2010 Pearson Education, Inc.
The displacement of an object in SHM is given by:
Example continued Copyright © 2010 Pearson Education, Inc.
13-6 The pendulum Copyright © 2010 Pearson Education, Inc.
A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).
The angle it makes with the vertical varies with time as a sine or cosine.
13-6 The pendulum Copyright © 2010 Pearson Education, Inc.
Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
13-6 The pendulum Copyright © 2010 Pearson Education, Inc.
However, for small angles, sin θ and θ are approximately equal.
13-6 The pendulum Copyright © 2010 Pearson Education, Inc.
Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:
Example Copyright © 2010 Pearson Education, Inc.
A clock has a pendulum that performs one full swing every 1.0 sec. The object at the end of the string weighs 10.0 N. What is the length of the pendulum?
Example Copyright © 2010 Pearson Education, Inc.
The gravitational potential energy of a pendulum is U = mgy. Taking y = 0 at the lowest point of the swing, show that y = L(1-cosθ).
13-6 The pendulum Copyright © 2010 Pearson Education, Inc.
A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:
13-6 The pendulum Copyright © 2010 Pearson Education, Inc.
In this case, it can be shown that the period depends on the moment of inertia:
Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• Period: time required for a motion to go through a complete cycle
• Frequency: number of oscillations per unit time
• Angular frequency:
• Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• The amplitude is the maximum displacement from equilibrium.
• Position as a function of time:
• Velocity as a function of time:
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• Acceleration as a function of time:
• Period of a mass on a spring:
• Total energy in simple harmonic motion:
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• Potential energy as a function of time:
• Kinetic energy as a function of time:
• A simple pendulum with small amplitude exhibits simple harmonic motion
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• Period of a simple pendulum:
• Period of a physical pendulum:
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• Oscillations where there is a nonconservative force are called damped.
• Underdamped: the amplitude decreases exponentially with time:
• Critically damped: no oscillations; system relaxes back to equilibrium in minimum time
• Overdamped: also no oscillations, but slower than critical damping
Summary of Chapter 13 Copyright © 2010 Pearson Education, Inc.
• An oscillating system may be driven by an external force
• This force may replace energy lost to friction, or may cause the amplitude to increase greatly at resonance
• Resonance occurs when the driving frequency is equal to the natural frequency of the system
13-7 Damped oscillations Copyright © 2010 Pearson Education, Inc.
In most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed:
This causes the amplitude to decrease exponentially with time:
13-7 Damped oscillations Copyright © 2010 Pearson Education, Inc.
This exponential decrease is shown in the figure:
13-7 Damped oscillations Copyright © 2010 Pearson Education, Inc.
The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.
13-8 Driven oscillations and Resonance Copyright © 2010 Pearson Education, Inc.
An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.
13-8 Driven oscillations and Resonance Copyright © 2010 Pearson Education, Inc.
If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.