Question

The motion of a damped harmonic oscillator can be modelled by a function of the form \(d(t)=(A \sin k t) \times 0.4^{c t},\) where \(d\) represents the distance as a function of time, \(t,\) and \(A, k,\) and \(c\) are constants. a) If \(d(t)=f(t) g(t),\) identify the equations of the functions \(f(t)\) and \(g(t)\) and graph them on the same set of axes. b) Graph \(d(t)\) on the same set of axes.

Short answer

Identify \(f(t) = A \sin(k t)\) and \(g(t) = 0.4^{c t}\); plot both and then plot \(d(t)\) as a damped sine wave.

Step-by-step solution

Title - Identify the function components

Start by recognizing that we need to split the given function into two separate functions, as specified. Given the function: \[d(t) = (A \sin(k t)) \times 0.4^{c t}\], where \d(t) = A \sin(k t) \times 0.4^{c t}, we can identify 1. \ f(t) = A \sin(k t) 2. \ g(t) = 0.4^{c t}

Title - Plot f(t)

Now, plot the function \(f(t) = A \sin(k t)\). This function represents a sinusoidal wave with amplitude A and frequency determined by k. The sine wave oscillates above and below the t-axis.

Title - Plot g(t)

Next, plot the function \(g(t) = 0.4^{c t}\). This function is an exponential decay function given that the base (0.4) is less than 1. As t increases, \( g(t) \) will decrease towards zero.

Title - Combine the functions

To graph the function \(d(t)\) on the same axes as \(f(t)\) and \(g(t)\), understand that the overall function \(d(t) = f(t) \times g(t)\) will resemble the sine wave of \(f(t)\) but will diminish over time due to the exponential decay factor from \(g(t)\). Therefore, the peaks and troughs of the sine wave will shrink towards zero.

Title - Sketch the final graph

Graph \(d(t)\) by following the pattern of \(f(t) = A \sin(k t)\) and simultaneously applying the damping effect contributed by \(g(t) = 0.4^{c t}\). The result is a sinusoidal wave whose amplitude decreases over time, eventually approaching zero as time goes on.

Key concepts

Sinusoidal Function

The function, \(\f(t) = A \sin(kt)\), represents a sinusoidal wave. A sinusoidal wave oscillates smoothly and periodically. The variable A determines the amplitude of the wave, which is the maximum distance from the t-axis. The value of k affects the frequency, or how many cycles the wave completes within a unit time. This type of function is common in scenarios involving repetitive oscillations, such as sound waves and alternating current. Visualizing a sine wave helps understand periodic behaviors in various fields of science and engineering. To graph a sinusoidal function, mark its amplitude and frequency, then draw the smooth curves above and below the t-axis accordingly.

Exponential Decay

Exponential decay functions describe processes that decrease rapidly at first and then slow down over time. In the function \(g(t) = 0.4^{ct}\), the base of the exponent (0.4) is between 0 and 1, which ensures the function decays. The variable c determines the rate of decay; higher values of c result in faster decay. Exponential decay is common in real-world phenomena such as radioactive decay, cooling of objects, and depreciation of assets. To plot an exponential decay function, begin at the starting value where t=0 and illustrate how the function value declines quickly at first and then tapers off as t increases.

Graphing Composite Functions

Composite functions combine two or more functions into a single expression. In our damped harmonic oscillator, the composite function \(d(t) = \(A \sin(kt) \times 0.4^{ct}\)\) merges the sinusoidal wave \(\f(t)\) and the exponential decay \(\bg(t)\). To graph it, plot \(\f(t)\) and \(\bg(t)\) separately. The result, \(\f(t) \times \bg(t) = d(t)\), will have a sinusoidal shape whose amplitude decreases over time due to the decay factor. This visually portrays how the harmonic motion loses energy over time. Graphically, this resembles the sine wave getting 'dampened', with peak and trough heights reducing as t increases.

Harmonic Motion

Harmonic motion refers to periodic oscillations, like those of spring systems or pendulums. It is typically modeled using sinusoidal functions. In our problem, the harmonic motion is represented by \(f(t) = A \sin(kt)\). When undammed, it would continuously oscillate. However, damping factors like friction or resistance cause the oscillations to reduce over time. This damping is modeled using the exponential decay function \(\bg(t) = 0.4^{ct}\). The combination of these functions in \(\bd(t)\) shows how real-world oscillatory systems eventually come to rest, demonstrating the energy dissipation in practical systems.

Mathematical Modelling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena. The function \(d(t) = A \sin(kt) \times 0.4^{ct}\) models the behavior of a dampened harmonic oscillator. Each component: \(A, \sin(kt),\) and \(\bg(0.4^{ct})\), corresponds to a physical aspect of the system. Parameters like amplitude (A), frequency (k), and damping coefficient (c) can be adjusted to reflect different real-life scenarios accurately. This allows scientists and engineers to predict system behavior under various conditions. Effective modeling bridges theoretical math and practical applicational understanding, enabling controlled experiments and innovations.