Oscillatory Motion Serway Chap 15 Oscillatory Motion in

Oscillatory Motion Serway (Chap. 15)

Oscillatory Motion in the real world may not fit some of our earlier models (linear or circular motion, uniform acceleration). Many phenomena are repetitive or oscillatory. Example: Block and spring M

Equilibrium: no net force M The spring force is always directed back towards equilibrium. This leads to an oscillation of the block about the equilibrium position. M F = -kx M x For an ideal spring, the force is proportional to displacement. For this particular force behaviour, the oscillation is simple harmonic motion.

Simple Harmonic Motion x(t) t In Simple Harmonic Motion (SHM), the displacement is a sinusoidal function of time, e. g. , : or Questions: Is a bouncing ball described by SHM, even if it returns to the same height? Is it periodic motion?

x(t) t In general, Phase The quantity (wt + f) is called the phase, and is measured in radians. The cosine function traces out one complete cycle when the phase changes by 2 p radians. The phase is not a physical angle! Three constants specify the motion are: 1) Amplitude, A 2) Angular Frequency, w 3) Initial phase (or phase constant), f

SHM: x(t) A T t -A A is the maximum value of x (x ranges from +A to -A). f gives the initial position at t=0: x(0) = A cosf. w is related to the period T and the frequency f = 1/T : T (period) is the time for one complete cycle (seconds). Frequency f (cycles per second or hertz, Hz) is the number of complete cycles per unit time.

The period T of the motion is the time needed to complete one cycle: and so x(t) = A cos (wt + f) cos (f) = cos(2 p + f) x (T) = x(0) if w. T = 2 p radians (or 360°) Units: radians/second or s-1

Example 1 The block is displaced a distance x = 5 cm from its equilibrium position and released from rest at time t = 0. Its motion is SHM with period 2 seconds. Write the function x(t). Steps: 1) Sketch a graph. 2) Write x = A cos (wt + f). 3) Evaluate A, w, and f. M x

Example 2 The block is at its equilibrium position and is set in motion by hitting it (and giving it an initial velocity) at time t = 0. Its motion is SHM with amplitude 5 cm and period 2 seconds. Write the function x(t). v 0 M x

QUIZ At t=0, a block is at x 0 = +5 cm, with positive velocity v 0. Its motion is SHM with amplitude 10 cm and period 2 seconds. If x(t) = A cos (wt + f), the phase constant f should be: A) B) C) D) E) 0 o 30 o 60 o -30 o -60 o M x 0 v 0

QUIZ A block on the end of a spring is pulled to position x=A and released from rest. In one full cycle of its motion, through what total distance does it travel? A) ½ A B) A C) 2 A D) 4 A M A