# Chapter 13 Oscillations about Equilibrium

## Chapter 13 Oscillations about Equilibrium

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Units of Chapter 13 The Pendulum Damped Oscillations Driven Oscillations in addition to Resonance 13-1 Periodic Motion Period: time required as long as one cycle of periodic motion Frequency: number of oscillations per unit time This unit is called the Hertz: 13-2 Simple Harmonic Motion A spring exerts a restoring as long as ce that is proportional to the displacement from equilibrium:

13-2 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. 13-2 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: It is then straight as long as ward 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 Uni as long as m Circular Motion in addition to Simple Harmonic Motion An object in simple harmonic motion has the same motion as one component of an object in uni as long as m circular motion:

13-3 Connections between Uni as long as m Circular Motion in addition to Simple Harmonic Motion 13-3 Connections between Uni as long as m Circular Motion in addition to Simple Harmonic Motion The position as a function of time: The angular frequency: 13-3 Connections between Uni as long as m Circular Motion in addition to Simple Harmonic Motion 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 Since the as long as ce on a mass on a spring is proportional to the displacement, in addition to also to the acceleration, we find that . Substituting the time dependencies of a in addition to x gives 13-4 The Period of a Mass on a Spring There as long as e, the period is 13-5 Energy Conservation in Oscillatory Motion In an ideal system with no nonconservative as long as ces, the total mechanical energy is conserved. For a mass on a spring: Since we know the position in addition to velocity as functions of time, we can find the maximum kinetic in addition to potential energies:

13-5 Energy Conservation in Oscillatory Motion As a function of time, So the total energy is constant; as the kinetic energy increases, the potential energy decreases, in addition to vice versa. 13-5 Energy Conservation in Oscillatory Motion This diagram shows how the energy trans as long as ms from potential to kinetic in addition to back, while the total energy remains the same. 13-6 The Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L ( in addition to negligible mass). The angle it makes with the vertical varies with time as a sine or cosine.

13-6 The Pendulum Looking at the as long as ces on the pendulum bob, we see that the restoring as long as ce is proportional to sin , whereas the restoring as long as ce as long as a spring is proportional to the displacement (which is in this case). 13-6 The Pendulum However, as long as small angles, sin in addition to are approximately equal. 13-6 The Pendulum Substituting as long as sin allows us to treat the pendulum in a mathematically identical way to the mass on a spring. There as long as e, we find that the period of a pendulum depends only on the length of the string:

13-6 The Pendulum 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 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, 13-7 Damped Oscillations In most physical situations, there is a nonconservative as long as ce of some sort, which will tend to decrease the amplitude of the oscillation, in addition to which is typically proportional to the speed: This causes the amplitude to decrease exponentially with time:

13-7 Damped Oscillations This exponential decrease is shown in the figure: 13-7 Damped Oscillations The previous image shows a system that is underdamped  it goes through multiple oscillations be as long as e coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating in addition to 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 in addition to Resonance An oscillation can be driven by an oscillating driving as long as ce; the frequency of the driving as long as ce may or may not be the same as the natural frequency of the system.

13-8 Driven Oscillations in addition to Resonance 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. Summary of Chapter 13 Period: time required as long as a motion to go through a complete cycle Frequency: number of oscillations per unit time Angular frequency: Simple harmonic motion occurs when the restoring as long as ce is proportional to the displacement from equilibrium. Summary of Chapter 13 The amplitude is the maximum displacement from equilibrium. Position as a function of time: Velocity as a function of time:

Summary of Chapter 13 Oscillations where there is a nonconservative as long as ce 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 An oscillating system may be driven by an external as long as ce This as long as ce 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

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