Problem 114 - A Graphical Approach to Precalculus with Limits

Question

The distance or displacement \(y\) of a weight attached to an oscillating spring from its natural position is modeled by \(y=4 \cos 2 \pi t,\) where \(t\) is time in seconds. Potential energy is the energy of position and is given by \(P=k y^{2},\) where \(k\) is a constant. The weight has the greatest potential energy when the spring is stretched the most. (a) Write \(P\) in terms of the cosine function. (b) Use an identity to write \(P\) in terms of \(\sin 2 \pi t\).

Step-by-Step Solution

Answer

(a) \( P = 16k \cos^2 2\pi t \); (b) \( P = 16k (1 - \sin^2 2\pi t) \).

Step 1: Express Displacement in Terms of Potential Energy

Given the formula for potential energy, \( P = k y^2 \), and displacement \( y = 4 \cos 2 \pi t \), substitute \( y \) into the potential energy equation to express \( P \):\[ P = k (4 \cos 2 \pi t)^2 \]Simplify the expression:\[ P = k \cdot 16 \cos^2 2 \pi t \]Thus, \( P \) in terms of the cosine function is \( 16k \cos^2 2 \pi t \).

Step 2: Use Trigonometric Identity to Rewrite P

Use the trigonometric identity for cosine squared to express \( P \) in terms of sine:\[ \cos^2 \theta = \frac{1+\cos 2\theta}{2} \]Substitute \( \theta = 2\pi t \):\[ \cos^2 2\pi t = \frac{1 + \cos 4\pi t}{2} \]Substitute this back into the potential energy expression:\[ P = 16k \cdot \cos^2 2\pi t = 16k \cdot \frac{1 + \cos 4\pi t}{2} \]Simplify:\[ P = 8k (1 + \cos 4\pi t) \]Utilize the identity \( \cos 4\pi t = 1 - 2\sin^2 2\pi t \):\[ P = 8k (1 + 1 - 2\sin^2 2\pi t) = 8k (2 - 2\sin^2 2\pi t) \]\[ P = 16k (1 - \sin^2 2\pi t) \]Therefore, \( P \) in terms of \( \sin 2\pi t \) is \( 16k (1 - \sin^2 2\pi t) \).

Key Concepts

Potential Energy

The concept of potential energy is essential in understanding systems like springs. Potential energy, often symbolized as \( P \), represents the energy stored in an object due to its position relative to other objects. In our case, it's about the energy in a spring when it's stretched or compressed. The general formula for potential energy in a spring-like system is given by:

• \( P = k y^2 \)

Here, \( k \) is a constant that represents the stiffness of the spring, and \( y \) is the displacement or how far the spring is stretched from its natural position. This equation tells us that potential energy increases as the square of the displacement. So, if you double the stretch, the energy increases by four times, showing the exponential nature of this energy type. This energy reaches its maximum when the spring is stretched or compressed to its extremes.

Trigonometric Identities

Trigonometric identities are mathematical equations that hold true for all values within their domains. They are incredibly useful, especially in simplifying expressions involving trigonometric functions like sine and cosine. In this exercise, we look at a specific identity to help us rewrite expressions:

• \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \)

This identity allows us to express \( \cos^2 \) as a function of \( \cos \) with double the angle, which can then be translated further using other identities if needed. This is particularly handy when handling oscillating systems such as springs, where expressing potential energy in different forms can simplify calculations or reveal new insights. Here, the identity helps to first express potential energy in terms of cosine and later in terms of sine, showing the flexibility and utility of these identities.

Harmonic Motion

Harmonic motion is a type of natural motion common to many systems, like springs, that can move back and forth in a regular pattern. In this exercise, we deal with harmonic motion as described by the cosine function \( y = 4 \cos 2 \pi t \). This function models the displacement \( y \) of a weight attached to a spring over time \( t \).In harmonic motion:

• The motion is periodic, meaning it repeats itself at regular intervals.
• The function describing the motion can often be expressed as either a sine or a cosine function.

The frequency of the motion refers to how often it completes a full cycle per unit of time. For this example, \( 2 \pi t \) indicates that the motion has a period of one second, because the cosine function completes a full cycle when its argument changes by \( 2\pi \). This regular pattern is key to understanding how energy, like potential energy, varies over time in systems displaying harmonic motion. Harmonic motion is fundamental in physics and engineering for analyzing systems in equilibrium and simple oscillations.