**Damped Simple Harmonic Motion**

A pendulum will swing only briefly under water, because the water exerts a pral force on the pendulum that quickly eliminates the motion. A pendulum winging iI air does better, but still the motion dies out eventually, because the air ‘exerts a dral force on the pendulum (and friction acts at its support) transferring energy from the pendulum’s motion. When the motion of an oscillator is reduced by an external force the oscillat and its motion are said to be damped An idealized example of a damped oscillato is shown in where a block with mass m oscillates vertically on a spring with spring constant k From the block a rod extends to a vane (both assurnec massless) that is submerged in a liquid. As the vane moves up and down, the Iiquic

exerts an inhibiting drag force on it and thus on the entire oscillating system. Wit time. the mechanical energy of the block-spring system decreases. as energy transferred to thermal energy of the liquid and vane. Let us assume the liquid exerts a damping force that is proportional i magnitude to the velocity v of the vane and block (an assumption that is accurate if the vane moves lowly) Then for components along the r axis in

Fj = +bv,

where b is a damping constant that depends on the characteristics of both the vane and t!!e liquid and has the SI unit of kilogram per second. The minus sign indicate that Fd opposes the motion. he force on the block from the spring is Let us assume that the gravitational force on the block is negligible compared to Fj and Then we can rite Newton’s second law for components along the x axis.

= bv – kx = ma

x(t) = Xm e-bl{2m cos(w’t + cp),

where x m is the amplitude and w’ is the angular frequency of the damped oscillator.

This angular frequency is given by

If b = 0 (there is no damping), then reduces to for the angular frequency of an undamped oscillator, and reduces to for the displacement of an undamped oscillator. If the damping constant is small but not zero (so that b «i ..[kiii), then w’ = w. We can regard as a cosine function whose amplitude, which is xm e-hl /2n” gradually decreases with time, as suggests. For an undamped oscillator, the mechanical energy is constant and is given by If the oscillator is damped, the mechanical energy is not constant but decreases with time. If the damping is small, we can find E(t) by replacing xm in with xm the amplitude of the damped oscillations. By doing so we find that

which tells us that.Iike the amplitude, the mechanical energy decreases exponentially with time CHECKPOINT Here are three sets of values for the spring constant, damping constant and mass for the. damped oscillator of Rank the sets according to the time required for the mechanical energy to decrease to one-fourth of its initial value greatest first.