1. 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves

2. A C B
equilibrium

3. 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.

4. 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.

5. 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

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

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

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