Vibrations

Introduction

In this module, we will study objects undergoing simple harmonic motion. When an object undergoes simple harmonic motion, there is a relationship between the displacement of the object and the amount of force being applied to it. A spring with a mass attached to it is an example of an object that will undergo simple harmonic motion. The force depends on the distance the spring is from its rest position and the type of spring it is. Pendulums approximate simple harmonic motion at small angles. Calculations using simple harmonic concepts will give a very good answer until the angle becomes relatively large.

Essential Questions

What does the graph of an object undergoing simple harmonic motion look like and how can amplitude, period and frequency be determined from that graph?
How can an expression for the displacement in simple harmonic motion be written in the forms $LaTeX: A\sin\omega t$ or $LaTeX: A\cos\omega t$ ?
How can an expression for velocity as a function of time be found for an object undergoing simple harmonic motion?
For an object undergoing simple harmonic motion, what is the relationship between acceleration, velocity, and displacement; and how do we determine when these values are at zero or at their greatest value?
What is the relationship between the frequency and period of an object undergoing simple harmonic motion?
What is the frequency and period of a system that obeys the differential equation form $LaTeX: \frac{d_2x}{df_2}=-\omega_2x$ ?
How does the total energy of an oscillating system depend on the amplitude of the motion and how can the kinetic energy and potential energy be graphed as a function of time?
How can the kinetic energy and potential energy of an oscillating system be calculated as a function of time?
How can the maximum displacement or velocity of a particle moving in simple harmonic motion be calculated when given the initial position and velocity?
When will a system resonate in response to a sinusoidal external force?
How can the period of oscillation of a mass on a spring be derived?
How can the expression for the period of a mass on a spring be applied?
How can problems be solved for a mass hanging from a spring and oscillating vertically?
How can problems be solved for a mass attached to a spring and oscillating horizontally?
How can the period of oscillation be determined for systems of springs in series or parallel?
How can the expression for the period of simple pendulum motion be derived?
How can the expression for the period of a simple pendulum be applied?
What approximations must be made when deriving the period of a pendulum?
How can the motion of a pendulum be used to determine the period of small oscillations?

Key Terms

Simple Harmonic Motion - Is a type of periodic motion where the restoring force is directly proportional to the displacement.
Pendulum - Is a weight suspended from a pivot so that it can swing freely.
Hooke's Law - In mechanics and physics , Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it.

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