The Pendulum

(4.48)

(4.49)

In other words, the period and frequency of a simple pendulum depend only on the length of the string and the acceleration due to gravity.

Damped Oscillations

In many real systems, nonconservative forces, such as friction, retard the motion. Consequently, the mechanical energy of the system diminishes in time, and the motion is said to be damped.

where b is a constant called the damping coefficient.

(4.50)

Figure 4.18 One example of a damped oscillator is an object attached to a spring and submersed in a viscous liquid.

Damped Oscillations

When the retarding force is small, the oscillatory character of the motion is preserved but the amplitude decreases in time, with the result that the motion ultimately ceases.

Forced Oscillations

At resonance the applied force is resonance in phase with the velocity and the

power transferred to the oscillator is a maximum.

Forced Oscillations

(a)In 1940 turbulent winds set up torsional vibrations in the Tacoma Narrows Bridge, causing it to oscillate at a frequency near one of the natural frequencies of the bridge structure.

(b)Once established, this resonance condition led to the bridge’s collapse.

Energy of the Simple

Harmonic Oscillator

(4.53)

That is, the total mechanical energy of a simple harmonic oscillator is a constant of the motion and is proportional to the square of the amplitude.

(4.54)