Problem 115 - A Graphical Approach to Precalculus with Limits

Question

Two types of mechanical energy are kinetic energy and potential energy. Kinetic energy is the energy of motion, and potential energy is the energy of position. A stretched spring has potential energy, which is converted to kinetic energy when the spring is released. If the potential energy of a weight attached to a spring is $$P(t)=k \cos ^{2} 4 \pi t$$ where \(k\) is a constant and \(t\) is time in seconds, then its kinetic energy is given by $$K(t)=k \sin ^{2} 4 \pi t$$ The total mechanical energy \(E\) is given by the equation \(E(t)=P(t)+K(t).\) (a) If \(k=2,\) graph \(P, K,\) and \(E\) in the window \([0,0.5]\) by \([-1,3],\) with \(\mathrm{Xscl}=0.25\) and \(\mathrm{Yscl}=1 .\) Interpret the graph. (b) Make a table of \(K, P,\) and \(E,\) starting at \(t=0\) and incrementing by \(0.05 .\) Interpret the results. (c) Use a fundamental identity to derive a simplified expression for \(E(t)\)

Step-by-Step Solution

Answer

Total energy \(E(t)\) is constant at 2 over the interval \([0, 0.5]\).

Step 1: Graph the Functions

First, we need to graph the functions \(P(t) = 2\cos^2(4\pi t)\), \(K(t) = 2\sin^2(4\pi t)\), and \(E(t) = P(t) + K(t)\) in the interval \([0, 0.5]\) with y-axis on \([-1, 3]\). For \(Xscl=0.25\) and \(Yscl=1\), plot the functions. Interpret the graph: both potential and kinetic energy oscillate with time, and their sum (total energy) remains constant at 2 due to conservation of energy.

Step 2: Create and Interpret the Table

Create a table where time \(t\) varies from 0 to 0.5 in increments of 0.05. For each time, calculate \(P(t)\), \(K(t)\), and \(E(t)\). Notice that \(P(t) + K(t)\) equals 2 for each entry, reiterating the conservation of mechanical energy. This constant energy demonstrates how potential and kinetic energies convert into one another.

Step 3: Simplify Expression for Total Energy

Recognize the trigonometric identity: \(\cos^2(\theta) + \sin^2(\theta) = 1\). Use this to simplify \(E(t) = 2(\cos^2(4\pi t) + \sin^2(4\pi t)) = 2\). The expression for total energy \(E(t)\) simplifies to 2, confirming the conservation of energy.

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics, representing the energy that an object possesses due to its motion. Whenever an object is in motion, it carries kinetic energy. The formula for calculating kinetic energy is given by \[ KE = \frac{1}{2} mv^2 \]where \( m \) stands for mass and \( v \) is velocity.

• An object at rest has zero kinetic energy.
• As the object's speed increases, its kinetic energy increases.

In the given problem, the kinetic energy function is represented as \( K(t) = k \sin^{2}(4\pi t) \). As time \( t \) changes, the sine function oscillates, causing \( K(t) \) to vary between 0 and \( k \). The graph visualizes this oscillation showing how energy is transformed over time.

Potential Energy

Potential energy is essentially stored energy, determined by an object's position or configuration. For instance, lifting an object against gravity "stores" energy, which we call gravitational potential energy. The formula for gravitational potential energy is \[ PE = mgh \]where \( m \) is mass, \( g \) is gravitational acceleration, and \( h \) is height.

• Potential energy can be in forms other than gravitational, such as elastic potential energy in a stretched spring.

In this exercise, potential energy is modeled by \( P(t) = k \cos^{2}(4\pi t) \). It varies with time as the spring system's position changes. The periodic nature of the cosine function results in energy shifting between kinetic and potential forms continuously.

Conservation of Energy

Conservation of energy is a core principle in physics stating that energy cannot be created or destroyed, only transformed from one form to another. This laws guarantees that the total mechanical energy (sum of kinetic and potential energy) in a closed system remains constant.

• This principle ensures that all energy types within a system interchangeably convert without loss.

The exercise beautifully illustrates conservation of energy through the relationship \( E(t) = P(t) + K(t) \), where the total mechanical energy \( E(t) \), consistently remains 2. Using the trigonometric identity \( \cos^{2}(\theta) + \sin^{2}(\theta) = 1 \), the simplification of \( E(t) = 2 \) conclusively demonstrates how potential and kinetic energy are interdependent, keeping total energy conserved and unaltered through the oscillation.