Sullivan Algebra and Trigonometry: Section 9.5 Objectives of this Section Find an Equation for an Object in Simple Harmonic Motion Analyze Simple Harmonic Motion Analyze an Object in Damped Motion Graph the Sum of Two Functions

The amplitude of vibration is the distance from the equilibrium position to its point of greatest displacement (A or C). The period of a vibrating object is the time required to complete one vibration - that is, the time required to go from point A through B to C and back to A.

Simple harmonic motion is a special kind of vibrational motion in which the acceleration a of the object is directly proportional to the negative of its displacement d from its rest position. That is, a = -kd, k > 0.

Simple Harmonic Motion An object that moves on a coordinate axis so that its distance d from the origin at time t is given by either

The frequency f of an object in simple harmonic motion is the number of oscillations per unit of time. Thus,

Suppose an object is attached to a pendulum and is pulled a distance 7 meters from its rest position and then released. If the time for one oscillation is 4 seconds, write an equation that relates the distance d of the object from its rest position after time t (in seconds). Assume no friction.

Suppose that the distance d (in centimeters) an object travels in time t (in seconds) satisfies the equation (a) Describe the motion of the object. Simple harmonic (b) What is the maximum displacement from its resting position?

Suppose that the distance d (in centimeters) an object travels in time t (in seconds) satisfies the equation (c) What is the time required for one oscillation? (d) What is the frequency?

Damped Motion The displacement d of an oscillating object from its at rest position at time t is given by where b is a damping factor (damping coefficient) and m is the mass of the oscillating object.

Suppose a simple pendulum with a bob of mass 8 grams and a damping factor of 0.7 grams/second is pulled 15 centimeters to the right of its rest position and released. The period of the pendulum without the damping effect is 4 seconds. (a) Find an equation that describes the position of the pendulum bob.

(b) Using a graphing utility, graph the function. (c) Determine the maximum displacement of the bob after the first oscillation.