Problem 43 - Algebra and Trigonometry

Question

An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=30, \quad a=18, \quad b=0.6, \quad T=4 $$

Step-by-Step Solution

Answer

The displacement function is \(d(t) = 18 e^{-0.01 t} \, \cos \left( \frac{\pi t}{2} \right)\). Plot it for \(t = 0\) to \(t = 20\).

Step 1 - Define the Displacement Function

For a damped harmonic oscillator, the displacement function can be given by:\[ d(t) = A e^{-bt/2m} \, \cos \left( \frac{2 \pi t}{T} \right) \]where \( A \) is the initial amplitude, \( e^{-bt/2m} \) is the damping factor, \( \cos \left( \frac{2 \pi t}{T} \right) \) represents periodic oscillation.

Step 2 - Plug in Given Values

Using the given values, \(m = 30 \) grams, \(a = 18 \) centimeters, \(b = 0.6 \) grams per second, \( T = 4 \) seconds, the displacement function can be written as:\[ d(t) = 18 e^{-(0.6/60) t} \, \cos \left( \frac{2 \pi t}{4} \right) \]

Step 3 - Simplify the Exponential Term

Simplify the exponential term:\[ e^{-(0.6/60) t} = e^{-0.01 t} \]Thus, the displacement function simplifies to: \[ d(t) = 18 e^{-0.01 t} \, \cos \left( \frac{\pi t}{2} \right) \]

Step 4 - Write the Final Displacement Function

The final function relating the displacement \(d\) of the object from its rest position after \(t\) seconds is: \[ d(t) = 18 e^{-0.01 t} \, \cos \left( \frac{\pi t}{2} \right) \]

Step 5 - Graph the Function

Plot the displacement function \(d(t) = 18 e^{-0.01 t} \, \cos \left( \frac{\pi t}{2} \right)\) for 5 oscillations using a graphing utility. Set the domain from \(t = 0\) to \(t = 20\) seconds (since the period \(T = 4\) seconds and 5 oscillations will cover \(5T = 20\) seconds).

Key Concepts

displacement function

To understand the behavior of a damped harmonic oscillator, one of the key elements we need is the 'displacement function'. This function tells us the position of the object at any time 't'. For a damped harmonic oscillator, the displacement function can be written as:

\[ d(t) = A e^{-bt/2m} \, \cos \left( \frac{2 \pi t}{T} \right) \] Where:

• \( A \) is the initial amplitude or the maximum initial displacement.
• \( e^{-bt/2m} \) is the damping factor, which shows how the amplitude decreases over time.
• \( \cos \left( \frac{2 \pi t}{T} \right) \) represents the periodic oscillation with respect to time.

This equation combines components of exponential decay and periodic oscillation to describe the motion accurately.

damping factor

The concept of 'damping factor' is crucial in understanding a damped harmonic oscillator. The damping factor is represented by the term \[ e^{-bt/2m} \]

• \( e \) is the base of the natural logarithm.
• \( b \) is the damping coefficient, measured in grams per second in this case.
• \( m \) is the mass of the object, measured in grams.
• \( t \) is time, measured in seconds.

In our specific exercise, the damping factor simplifies to \ e^{-0.01t} \, indicating that the amplitude of the motion decreases exponentially over time.

harmonic motion

Harmonic motion refers to the periodic oscillation of the object around its equilibrium position. When undamped, this motion would be purely sinusoidal, described by a function like: \[ A \cos \left( \frac{2 \pi t}{T} \right) \] However, in the case of a damped harmonic oscillator, we have the periodic component: \[ \cos \left( \frac{2 \pi t}{T} \right) \]

• \( T \) is the period of the oscillation, which is the time it takes to complete one full cycle.
• \( t \) is the time.
• The cosine function ensures that the displacement oscillates back and forth.

Combined with the damping factor, it describes a motion that gradually comes to a stop, rather than continuing endlessly.

exponential decay

Exponential decay shows how the amplitude of the oscillation decreases over time. In the displacement function, this is represented by: \[ e^{-bt/2m} \]This exponential term ensures that the displacement becomes smaller as time increases. Effectively, it 'damps' the motion, preventing it from oscillating forever. For our particular problem, the exponential decay term simplifies to: \[ e^{-0.01 t} \]This tells us that as time progresses, the initial amplitude of 18 cm reduces exponentially, making each successive oscillation smaller.

periodic oscillation

Periodic oscillation in a damped harmonic oscillator is given by: \[ \cos \left( \frac{2 \pi t}{T} \right) \]

• The cosine function repeats its values in regular intervals, which is called periodicity.
• \( T \) is the period, the duration of one full cycle of the oscillation.

In our specific case, the oscillator has a period of 4 seconds. Combining periodic oscillation with exponential decay gives us a motion where the object oscillates back and forth but with decreasing amplitude. This results in a graph that shows diminishing peaks with regular intervals, effectively capturing the essence of damped harmonic motion.