Problem 68 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

The motion of a spring can be modeled by \(y=A \cos k t\), where \(y\) is the vertical displacement (in feet) of the spring relative to its position at rest, \(A\) is the initial displacement (in feet), \(k\) is a constant that measures the elasticity of the spring, and \(t\) is the time (in seconds). a. You have a spring whose motion can be modeled by the function \(y=0.2 \cos 6 t\). Find the initial displacement and the period of the spring. Then graph the function. b. When a damping force is applied to the spring, the motion of the spring can be modeled by the function \(y=0.2 e^{-4.5 t} \cos 4 t\). Graph this function. What effect does damping have on the motion?

Step-by-Step Solution

Answer

For the given spring function \(y=0.2 \cos 6 t\), the initial displacement is 0.2 feet, and the period is \(\pi /3\) seconds. For the damped spring function \(y=0.2 e^{-4.5 t} \cos 4 t\), there is a reduction in amplitude over time due to damping.

Step 1: Identify initial displacement and period from the first equation

From \(y = 0.2 cos 6t\), it determined that the initial displacement, \(A = 0.2\) feet. The value of \(k\) in the equation is 6, so the period of motion can be found using the formula \(T = 2\pi/k = 2 * \pi /6 = \pi /3\) seconds.

Step 2: Illustrate the function

Now, plot \(y = 0.2 cos 6t\). On the horizontal axis, the time \(t\) is marked. On the vertical axis, the spring displacement \(y\) is marked. The graph starts at a peak because of the positive cosine function, the maximum displacement is 0.2 (the amplitude), and the function completes a full cycle every \(\pi /3\) seconds (the period).

Step 3: Analyze and illustrate the damped function

Next, plot the damped function \(y = 0.2 e^{-4.5t} cos 4t\). This also represents the spring's movement, but with damping over time. On the graph, we see that the amplitude of oscillation declines over time, representing the effect of the damping force.

Step 4: Observe the effect of damping on the motion

The damping makes the spring comes to rest progressively faster. In other words, the amplitude of the spring oscillation decreases as time progresses. This progressive reduction in amplitude is called 'damping'.

Key Concepts

Spring Constant

In harmonic motion, the spring constant, often represented by the symbol \( k \), is a crucial element. It represents how stiff or flexible a spring is. The spring constant is defined as the ratio of the force affecting the spring to the displacement caused by it. The higher the spring constant, the stiffer the spring. Conversely, a lower spring constant means a more flexible spring.

This constant is vital when studying harmonic motion because it dictates the frequency at which the spring will naturally oscillate. For example, in the equation \( y = A \cos(kt) \), \( k \) controls how fast the cosine function cycles. Therefore, knowing the spring constant allows us to calculate other important properties like the period of the motion, using the formula \( T = \frac{2\pi}{k} \), where \( T \) is the period.

Damping Force

Damping force is an essential concept in harmonic motion, particularly when describing real-world systems, such as how a spring eventually stops moving. Damping force is any force that dissipates the energy of the system and thereby reduces the amplitude of oscillations over time.

This force comes from energy losses due to various factors like friction or air resistance. When damping is introduced, the motion of a spring changes. For instance, in the equation \( y = 0.2 e^{-4.5 t} \cos(4t) \), the exponential decay factor \( e^{-4.5 t} \) is causing the amplitude of motion to diminish over time.

This results in the spring coming to rest faster as compared to when no damping is present. Understanding the damping effect helps us design systems that can control vibrations effectively, such as in vehicle suspensions or building shock absorbers.

Amplitude

Amplitude in the context of harmonic motion refers to the maximum extent of displacement from the resting position. It is represented by the variable \( A \) in trigonometric motion equations like \( y = A \cos(kt) \).

The amplitude tells us the furthest point reached by the oscillating object, either in the positive or negative direction from its rest position. When you analyze equations like \( y = 0.2 \cos(6t) \), the amplitude \( A \) is 0.2 feet, illustrating the maximum displacement the spring achieves from its resting state.

• Large amplitude implies larger energy in the oscillations.
• In damped systems, amplitude decreases over time.

Understanding amplitude is crucial as it can indicate how much energy the harmonic system is carrying.

Periodic Function

A periodic function is one that repeats its values at regular intervals, known as periods. In harmonic motion, this concept is best demonstrated with trigonometric functions like sine and cosine.

For the equation \( y = 0.2 \cos(6t) \), the periodic function here is the cosine function that repeats after every full cycle. The period \( T \) of a periodic function, calculated as \( T = \frac{2\pi}{k} \), describes the duration of one complete cycle. For the given equation, \( T \) is \( \pi / 3 \).

Periodic functions are integral to understanding motion systems that rely on repetitiveness, such as pendulums, springs, or waves. They provide insight into predictable patterns and frequencies that are crucial in fields like engineering, physics, and even music.