1.) We want to find the % Energy LOST, if any, when one time constant has elapsed:

Theory: Based on the decay function that E(t) represents for a damped function, the aforementioned graph, and the fact that Energy is not conserved in a damped oscillator, there should be at least some amount of energy lost given the passing of any length of time. (Intuitively, this automatically eliminates choice D as being our answer)

2.) Applying the proportionality of Amplitude and Energy, we get...

A. LOST 63% of its initial energy

B. LOST 37% of its initial energy

C. LOST half of its initial energy

D. lost NO energy

E. LOST 13% of its initial energy

What do these equations for E(t) represent?

As seen/mentioned both in our text and in Reading Assignment Question #4, the equation for damped oscillation is the same as harmonic oscillation, just with the added e^-bt/2m to account for the dampening. This addition to the original harmonic equation turns it into an exponential decay function, representing the loss of energy, E, over time, t, as a frictional force acts against the direction of the oscillator with a damping constant, b , causing the energy of the system to be lost as thermal energy.

The Question

``READ THIS QUESTION CAREFULLY: For non-ideal oscillators (such as a real pendulum) energy is lost and the amplitude (which is the maximum displacement) is no longer constant but also decreases with time. How fast the energy lost is described by a TIME CONSTANT,

T

= m/b. (Compare this idea to equation 13-75: the exponent must be dimensionless, hence m/b must have the unit of time.) Typically after ONE time constant has elapsed, the system has ...(hint: this is when t =

T

)``

**Relating Energy and Amplitude in Damped Oscillations: An Analysis of Reading Assignment 4, Question #3**

The Related Topic

This question is relating what we have learnt in our prereading on the Energy of a damped oscillation being proportional to the amplitude, where amplitude equals:

How does the Energy relate to the Amplitude (max position) of a damped oscillator?

(source: Physics for Scientists and Engineers An Interactive Approach UBC Custom Volume 1 p.366 Figure 13-32)

Your Choices...

NOTE: as mentioned in the question, this

is equation 13-75

The position of a damped oscillator is given by the equation:

Which, when graphed with known variables specific to a given damped oscillator, looks something along the lines of the graph to the left, given as an example in the text.

Notice, however, that as time progresses with a damped oscillator, the peaks of the position (amplitude) decrease. This decay of amplitude (represented by the red dotted lines) also represents the decay of the energy of the oscillations as they are being damped by a frictional force.

Using This Information to Solve the Problem

Application: Because we are told that

T

= m/b = t , we can substitute m/b for t in our amplitude equation, so that:

A(t) = Ae^(-bm/2bm) =>b and m cancel

=

Ae^(-1/2)

And, substituting this into our energy equation...

Given: A(t) =

Ae^(-1/2)

, we can substitute this into our equation for energy, where:

E(t) = 1/2 kA(t)^2

= 1/2k (Ae^(-1/2))^2 => e^-1/2 = 0.60653

= 1/2k(0.60653A)^2

=0.37 (1/2kA^2)

This shows that after one time period, there is 0.37, or 37%, of the initial energy*** REMAINING in the damped oscillator, meaning there has been a 63% LOSS of Energy (E(final)-E(initial) =37%-100%= -63% [negative sign means energy lost]), making our answer

A: LOST 63% of its original energy.

The answer could also have been derived in simpler steps by solely using the expression for Energy that is given in equation 13-75, which already integrates amplitude as a function of time, substituting m/b for t and solving using similar steps to those seen in the final step of this explanation. However, for the purposes of this learning object, I chose to separate the two equations - Amplitude and Energy - so as to better show the relationship between them and to highlight the importance of the value e^-bt/2m in differentiating between a harmonic oscillator and the decay function of a damped oscillator.

Note:

Key Ideas:

Energy is proportional to Amplitude (a decrease in amplitude means a loss in energy)

The decay trend of the Energy vs time graph of a damped oscillator can be related to the peaks of a position vs time graph, where the peaks represent the max position of the oscillator, and thus the amplitude

The Energy vs time graph of a damped oscillator is seen to be an exponential decay function attributed to the non-conservation of energy (energy is transformed to thermal energy via an opposing frictional force of a damping constant, b)

The difference between the equation for a harmonic oscillator (ideal system; energy is conserved) and a damped oscillator (non-ideal system; energy NOT conserved) is the addition of the decay variable of e^-bt/2m

***initial energy is when t=0, so E(0) = 1/2k(Ae^-b(0)/2m)^2 = 1/2kA^2, making this a constant in the given question and concerning ourselves solely with the coefficient of 0.37 (NOT 0.37/2, which some students might have been confused about) as being the amount of energy remaining in the damped oscillator