7.5 Simple Harmonic Motion; Damped Motion; Combining Waves

A C B equilibrium

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.

Theorem 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? A = |-15| = 15 centimeters.

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? Period : (d) What is the frequency? frequency oscillations per second.

Theorem 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.