586 CHAPTER 8 Applications of Trigonometric Functions Figure 47 Damped motion t a 2a Solution The function ( ) ( ) = d t t 10sin 5 is of the form ω ( ) ( ) = d t a t sin where = a 10 and ω = 5. (a) The motion is simple harmonic. (b) The maximum displacement of the object from its rest position is the amplitude: = a 10 meters. (c) The time required for one oscillation is the period: π ω π = = = T Period 2 2 5 seconds (d) The frequency is the reciprocal of the period. π = = = f T Frequency 1 5 2 oscillation per second Now Work PROBLEM 15 3 Analyze an Object in Damped Motion In the models discussed up to now, the motion was simple harmonic. That is, they assumed no force was retarding the motion. However, most physical phenomena are affected by friction or other resistive forces. These forces remove energy from a moving system and thereby damp its motion. For example, when a mass hanging from a spring is pulled down a distance a and released, the friction in the spring causes the distance the mass moves from its rest position to decrease over time. As a result, the amplitude of any real oscillating spring or swinging pendulum decreases with time due to air resistance, friction, or other forces. See Figure 47. A model that describes this phenomenon maintains a sinusoidal component, but the amplitude of this component decreases with time to account for the damping effect. Moreover, the period of the oscillating component is affected by the damping. The next theorem, from physics, describes damped motion. THEOREM Damped Motion The displacement d of an oscillating object from its rest position at time t is given by ω ( ) = − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ( ) − d t ae b m t cos 4 bt m2 2 2 2 where b is the damping factor or damping coefficient and m is the mass of the oscillating object. Here a is the displacement at = t 0, and π ω 2 is the period under simple harmonic motion (no damping).
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