The velocity vet) of a particle undergoing SHM is graphed

Chapter , Problem 2

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The velocity vet) of a particle undergoing SHM is graphed in Fig. I5-18b. Is the particle momentarily stationary, headed toward -x"" or headed toward +xlll at (a) point A on the graph and (b) point E? Is the particle at -XIII' at +XIII , at 0, between -XIII and 0, or between 0 and + XIII when its velocity is represented by (c) point A QUESTIONS 403 Pendulums Examples of devices that undergo simple harmonic motion are the torsion pendulum of Fig. 15-7, the simple pendulum of Fig. 15-9, and the physical pendulum of Fig. 15-10. Their periods of oscillation for small oscillations are, respectively, T= 27T~ (torsion pendulum), (15-23) T = 27Tvug (simple pendulum), (15-28) T = 27T VIImgh (physical pendulum). (15-29) Simple Harmonic Motion and Uniform Circular Motion Simple harmonic motion is the projection of uniform circular motion onto the diameter of the circle in which the circular motion occurs. Figure 15-13 shows that all parameters of circular motion (position, velocity, and acceleration) project to the corresponding values for simple harmonic motion. Damped Harmonic Motion The mechanical energy E in a real oscillating system decreases during the oscillations because external forces, such as a drag force, inhibit the oscillations and transfer mechanical energy to thermal energy. The real oscillator and its motion are then said to be damped. If the damping force is given by Fd = -bv, where j7 is the velocity of the oscillator and b is a damping constant, then the displacement of the oscillator is given by X(t) = Xm e-btl21ll cos( Wi t + ), (15-42) where Wi, the angular frequency of the damped oscillator, is given by Wi = k (15-43) m If the damping constant is small (b ~ Ykiii), then Wi = w, where w is the angular frequency of the undamped oscillator. For small b, the mechanical energy E of the oscillator is given by (15-44) Forced Oscillations and Resonance If an external driving force with angular frequency Wd acts on an oscillating system with natural angular frequency w, the system oscillates with angular frequency Wd' The velocity amplitude VIII of the system is greatest when (15-46) a condition called resonance. The amplitude Xm of the system is (approximately) greatest under the same condition. and (d) point E? Is the speed of the particle increasing or decreasing at (e) point A and (f) pointE?