The Physical Pendulum

In Fig. 15.11 an extended body is pivoted freely about an axis that does not pass through its center of mass. Such an arrangement forms a physical pendulum that executes simple harmonic motion for small angular displacements. A practical example of a physical pendulum is an arm or a leg. If d is the distance from the pivot to the center of mass, the restoring torque is -mgd sin θ (toward decreasing θ). The rotational form of Newton's second law, τ = Ia, is

where is the moment of inertia about the given axis. If we make the small-angle approximation, sin θ ~ θ, then

(15.15)

which is the equation for simple harmonic oscillation. Comparison with Eq. 15.5 shows that

(Physical pendulum) - (15.16)

and

(15.17)

If the location of the center of mass and d are known, then a measurement of the period allows us to determine the moment of inertia of the body.

FIGURE 15.11

7.4 DAMPED OSCILLATIONS (Optional)

Thus far we have ignored the inevitable energy losses that oc­cur in real situations. Such losses may arise from external fluid resistance or from "internal friction" within a system. The en­ergy, and consequently the amplitude, of such a damped oscil­lator decrease in time. To formulate the equation for damped oscillations, we consider the situation depicted in Fig. 15.13 which shows a block immersed in a liquid. When the velocity is low, the damping is due to a resistive force f that is proportional to the velocity (see Section 6.4):

F = – γ v (15.20)

where γ, measured in kg/s, is the damping constant. If we ignore the buoyancy of the fluid, Newton's second law applied to the block is

where x is the displacement from equilibrium. This equation may be written in the form

(15.21)

This form of differential equation arises in other mechanical or nonmechanical damped oscillations. Experience tells us that the mass will oscillate with ever-decreasing amplitude. As you can verify by substitution, the solution to Eq. 15.21 is

(15.22)

15.6 FORCED OSCILLATIONS (Optional)

The loss in energy of a damped oscillator may be compensated for by work done by an external agent. Newton's second law applied to such a forced or driven oscillator yields

(15.25)

When the force is first applied, the motion is complex. How­ever, the system ultimately settles into a steady-state oscilla­tion. At this stage the energy dissipated by the damping is ex­actly balanced by the external input. The steady-state solution to Eq. 15.25 is

(15.26)

where δ is the phase angle between the displacement x and the external force F. Notice that the amplitude is constant in time and that ω_е is the angular frequency of the external driving force. When Eq. 15.26 is substituted into Eq. 15.25, we are led to conclude (the details are omitted)

(15.27)

Each driving frequency is characterized by its own amplitude, as shown in Fig. 15.17. At w_e = 0, the amplitude is merely the static extension F0/mw_e = F0/k. As the external angular fre­quency ω_e is increased, the amplitude rises until it reaches a maximum at ω_max, which is somewhat below ω₀. At higher fre­quencies, the amplitude again decreases. Such a response is called resonance and ω_max is called the resonance angular fre­quency. When γ is small, the resonance curve is narrow and the peak occurs close to the natural angular frequency ω₀. For large γ, the resonance is broad and the peak is shifted to lower fre­quencies. The value of γ may be so large that there is no reso­nance. At the resonance frequency the external force and the velocity of the particle are in phase. As a result, the power transfer (P = F • v) to the oscillator has its maximum value. At frequencies above or below the resonance value, the force and velocity are not in phase, so the power transfer is lower.

CHAPTER 8

Mechanical Waves

Major Points

1. Wave characteristics; linear superposition; reflection and transmission of pulses.

2. The speed of a pulse on a string is determined by its tension and its mass density.

(a) Traveling harmonic waves.

(b) The distinction between particle velocity and wave velocity.

3. Standing waves. Resonant standing waves in strings.

4. Energy transport by a wave on a string.

5. The wave equation is a differential equation satisfied by traveling waves.

In previous chapters we studied the motion of particles, rigid bodies, and fluids. We now begin the study of wave motion. A wave is a disturbance that travels, or propagates, without the transport of matter.

Mechanical waves, such as water waves or sound waves, travel within, or on the surface of, a material with elastic properties: There must be some mechanism that tends to restore the medium to its normal or equilibrium state. In contrast. electromagnetic waves, such as light and TV signals, are nonmechanical; and can propagate through a vacuum.

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