2. Use physical and mathematical intuition to guess a solution As suggested in figure 3.1.3, the solution might be of the form
3.21 Drag force for turbulent flow. (a) It is easier to think about drag forces in a reference frame in which the object is stationary and the fluid medium
(air, water, etc.) is moving past it. In laminar flow, the fluid moves around the object with relatively little net disruption to its velocity. However, in turbulent flow, the velocity of the fluid is changed violently. Consider the volume of fluid that will impinge on the object in a timet, as shown in figure 3.P.2. This has a lengthvt, and a cross-sectional areaAequal to that of the object. Assume that a fractionCd/2 of the momentum of this volume of fluid is transferred to the object. (Cdis called the “drag coefficient.”) Show
Figure 3.P.2 In timet, all the air in the cylinder will hit the front of the spherical object.
that the resulting force on the object has a magnitude Fdrag = Cd 2 Av2ρ, whereρ is the density of the fluid.(b) For an airplane,Cd≈0.03. What power is required from the engines for an airplane withA= 15 m2 to fly at a speed of 150 kph near sea level, where the density of air is 1.2 kg/m3? (c)Repeat part (b) for a speed of 300 kph.
4 Driven Oscillations and Resonance
When you drive an oscillator at its resonant frequency Then the amplitude of the oscillation will become huge.
In the equation above, it becomes infinite, But in practice there will be some damping That prevents that.
You have known this since your childhood, This is how you swing on a swing.
If you live in a snowy climate, you know (or at least should know) that a trick To get your car out of a snow bank is To rock it back and forth—
If you get the frequency right you will make the car oscillate With a large amplitude
And dislodge it.
The electrical analogue is used to tune a radio.
From “The Driven Oscillator” (not originally intended to be a poem) by Professor B. Paul Padley, Rice University
4.1 Resonance
As we saw in chapter 3, the oscillations of a macroscopic oscillator decay over time because the energy leaks out into the surroundings. For an oscillation to be sustained, this energy loss must be balanced by the energy added to the oscillator. We see examples of this all the time. A child on a swing uses her legs to pump the amplitude higher and higher. The seat in a bus vibrates, because it is driven by the engine which is shaking in a regular way. A washing machine shakes, especially in the spin cycle; in this case, the energy is provided by the motor which spins the tub with the clothes in it. The current inside a radio receiver oscillates, driven by the radio waves broadcast by the transmitter. The straps on your backpack, and your hair (if it’s long enough) swing back and forth as you walk, driven by the energy you put into moving your legs (which also moves your body up and down a little with each step).
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The behavior of oscillators that are driven, including the above examples, is perhaps the single most important idea in all of physics. Essentially every area of physics has strong connections with driven oscillators, and many seemingly unrelated phenomena can be understood qualitatively by analogy with driven oscillators.
You have probably noticed that if an oscillator is driven with a periodic force of just the right frequency, the motion of the oscillator becomes very large, much larger than for other frequencies of drive force. For example, if you have an old car that rattles sometimes, you may have noticed that the rattle is worst when the engine is running at a particular speed—for lower speedsorfor higher speeds the rattle is less noticeable.
This phenomenon of a strong response at a certain frequency is called “resonance.” It can be annoying and even destructive, leading to failure of mechanical components, but it can also be used to almost magical effect in the design of ultrasensitive detectors, and in medical imaging, as shown in figure 4.1.1.
Following the approach of physics, we begin with the simplest possible driven oscillator: a mass on a spring. However, by now you recognize that this can represent a vast array of physical systems, including electronic circuits (which we’ll explore later in this chapter). We will also assume to start that the energy injected into the oscillator comes from the simplest possible periodic source: a sinusoidal driving force.
However, this is actually no restriction at all, since we will show thatanyperiodic driving force can be represented as a sum of sinusoids, and that the response of the linear
Figure 4.1.1 Magnetic resonance images of a human brain. (Image © Katrina Brown/Dreamstime.com)
Figure 4.1.2An unbalanced washing machine (top view).
oscillators that are our main focus is simply the sum of the responses to the individual sinusoids.
Furthermore, many real-world driving forcesdohave a simple sinusoidal form.
Many driving forces arise from a circular motion. For example, consider again the tub of a washing machine, in which the clothes are unevenly distributed. The position of the heaviest part of the clothes is shown by the dot in figure 4.1.2, and the angle of a line to this point relative to horizontal isθ. If the tub rotates at constant angular velocityω, then θ=ωt. The horizontal position of the bunch of clothes is thenx=rcosθ=rcosωt, so as the tub rotates, this creates a sinusoidal driving force in thex-direction.
For a mass hanging on a spring (figure 4.1.3), one way to apply the driving force is to move the support point for the spring sinusoidally. The force from the spring is clearly related to the motion ofx relative to xc, i.e.,
Fspring= −kx−xc
(e.g., ifxandxcare both shifted upward by the same amount, the spring force should be zero.) We move xc sinusoidally with amplitude Ad and angular frequency ωd, where subscript “d” indicates “drive.” For example, ifxc=Adcosωdt, then we get a sinusoidal driving force:
Fspring= −kx−xc= −kx+kAdcosωdt= − !kx
usual spring force
+F0cosωdt ! sinusoidal driving force
with
F0=kAd. (4.1.1)
Figure 4.1.3 A sinusoidal driving force can be applied by moving the support pointxcsinusoidally. The positionxof the mass is defined relative to the equilibrium position, as indicated by the short horizontal line next tox. Similarly the position of the support point, xc, is defined relative to its own equilibrium position, as indicated by the short horizontal line next toxc.
Before starting our quantitative analysis, let’s qualitatively consider what to expect. If the frequency of the drive is very low, then the mass should simply move up and down together with the support point, that is,x =xc =Adcosωdt. On the other hand, ifωdis very high, then the sign of the drive force keeps changing at a high frequency. So, the drive force doesn’t have much time to accelerate the mass before the sign of the drive force changes. So, for largeωd, we expect a small motion of the mass.
Somewhere between low ωd and high ωd we expect to find a resonant response, that is, a special frequency of drive force for which the mass oscillates with large amplitude.
The damped oscillator without a drive force is just a special case of the damped driven oscillator, with zero drive amplitude, that is, F0=0. Thus, thefull solution for the oscillator with both driving and damping should include the possibility of an oscillation at angular frequency ωv that decays away exponentially and has an amplitude and phase that depend on the initial position and velocity. However, for times well beyond t =0, it is reasonable to expect that the effect of these initial conditions will decay away, and that we will see a “steady-state” oscillation in which the energy per cycle delivered to the system by the drive equals the energy per cycle that leaks away from the system because of the damping. This “steady-state solution”
is actually the one of most interest, since it persists indefinitely.
We consider a general sinusoidal drive force
Fdrive=F0cosωdt, (4.1.2)
which might be applied by a motion of the support point (equation (4.1.1), or might be applied in some other way.
To find the behavior of the system,x(t), we again follow our three-step procedure: