Problem 44 - 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. $$ m=15, \quad a=16, \quad b=0.65, \quad T=5 $$

Step-by-Step Solution

Answer

The displacement function is \( d(t) = 16 e^{-0.02167 t} \text{cos} \bigg( \frac{2π}{5} t \bigg)\).

Step 1: Understanding the problem

An object of mass 15 grams is attached to a coiled spring with a damping factor of 0.65 grams per second. The object is initially displaced 16 cm from its rest position and released. We need the displacement function over time and its graph.

Step 2: Identify the necessary constants and formulas

Given:\(m = 15\text{ grams}\)\(a = 16\text{ cm}\)\(b = 0.65\text{ grams per second}\)\(T = 5\text{ seconds}\)The damping factor implies this is a damped harmonic motion problem.

Step 3: Write the displacement function for damped harmonic motion

The general form of the displacement function for damped harmonic motion is:\[d(t) = a e^{-\frac{b}{2m}t} \text{cos}\bigg( \frac{2\text{π}}{T} t \bigg)\]Substituting the given values:\(m = 15\), \(a = 16\), \(b = 0.65\), and \(T = 5\), we obtain:\[d(t) = 16 e^{-\frac{0.65}{2 \times 15} t} \text{cos} \bigg( \frac{2\text{π}}{5} t \bigg)\]

Step 4: Simplify the exponent term

Simplify \( \frac{0.65}{30} \):\( \frac{0.65}{30} = 0.02167\)Therefore, the function becomes:\[d(t) = 16 e^{-0.02167 t} \text{cos} \bigg( \frac{2π}{5} t \bigg)\]

Step 5: Graph the displacement function

Use a graphing utility to graph \( d(t) = 16 e^{-0.02167 t} \text{cos} \bigg( \frac{2π}{5} t \bigg) \) for 5 oscillations (which corresponds to 5 periods, or 25 seconds). Ensure the graph shows the damped oscillations.

Key Concepts

Displacement Function

Displacement in the context of harmonic motion refers to the position of a mass relative to its equilibrium point over time. For a damped harmonic oscillator, the displacement function combines aspects of both simple harmonic motion and an exponential decay, due to damping.

The general form is given by:
\( d(t) = a e^{-\frac{b}{2m}t} \text{cos} \bigg( \frac{2\text{π}}{T} t \bigg) \)
Here,

• \(a\) is the initial displacement (amplitude).
• \(b\) is the damping factor.
• \(m\) is the mass.
• \(T\) is the period of the oscillation under simple harmonic conditions.

In our example, using values \(m = 15\), \(a = 16\), \(b = 0.65\), and \(T = 5\), the specific function derived becomes:
\(d(t) = 16 e^{-0.02167 t} \text{cos} \bigg( \frac{2π}{5} t \bigg)\).
The exponential term \(e^{-0.02167 t}\) represents the damping effect, slowly reducing the amplitude of oscillation.

Damping Factor

The damping factor (\text b \text ) measures how quickly the oscillation's amplitude decreases over time. It results from forces like friction or air resistance that oppose the mass's motion. In our displacement function,
\(d(t) = 16 e^{-0.02167 t} \text{cos} \bigg( \frac{2π}{5} t \bigg)\),
\(e^{-0.02167 t}\) indicates that at each moment \(t\), the amplitude is scaled down by this term.
An increasing damping factor makes the oscillation die out more quickly while a smaller one allows the oscillation to persist longer, albeit still diminishing. Adjusting \(b\) changes how fast the exponential decay occurs, influencing the rate at which energy is lost from the system.

Oscillation Graphing

Graphing the oscillation helps to visualize the motion over time. You'll notice several features upon graphing:

• The amplitude decreases over time because of the damping factor \(e^{-0.02167 t}\).


• The oscillation frequency remains constant as dictated by the cosine term \(\big( \text{cos}\big(\frac{2π}{5} t\big)\big)\), representing undamped periodic motion.


• The envelope of the peaks (formed by the exponent) will show a decaying pattern.

To graph the function for 5 oscillations (or 25 seconds), you can use graphing utilities like Desmos, a TI graphing calculator, or any other similar tools. Plug in the function \(d(t) = 16 e^{-0.02167 t} \text{cos} \bigg( \frac{2π}{5} t \bigg)\) and adjust the time range from \(t = 0\) to \(t = 25\) seconds to view the first 5 oscillations.

Harmonic Motion Period

The period \(T\) of harmonic motion is the time it takes for one complete oscillation. It's calculated as the inverse of the frequency. In our case:

• The period \(T\) is \(5\) seconds, indicating that it takes 5 seconds for one full back-and-forth motion cycle.

For damped harmonic motion, the period primarily remains the same, but the apparent changes may occur due to damping, slowing the oscillation down. Regardless of damping, each cycle still theoretically completes in \(T\) seconds, though the amplitude gets notably smaller every cycle. This property helps in understanding the persistence of oscillations despite energy loss.

Simple Harmonic Motion

In simple harmonic motion, or SHM, the restoring force on the mass is proportional and opposite to its displacement, typically resulting in sinusoidal oscillations. Without any damping (\(b = 0\)), the displacement function simplifies:
\(d(t) = a \text{cos} \bigg( \frac{2π}{T} t \bigg)\)
From this, the mass oscillates indefinitely with constant amplitude and period.
However, in our scenario, damping is introduced, modifying the SHM into damped harmonic motion. The distinction lies in the presence of an exponential decay factor (\(e^{-\frac{b}{2m}t}\)), indicating how real-world frictions affect otherwise ideal oscillatory motions.