Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

If a mass that is attached to a spring is raised \(y_{0}\) feet and released with an initial vertical velocity of \(v_{0} \mathrm{ft} / \mathrm{sec}\), then the subsequent position \(y\) of the mass is given by $$ y=y_{0} \cos \omega t+\frac{v_{0}}{\omega} \sin \omega t, $$ where \(t\) is time in seconds and \(\omega\) is a positive constant. (a) If \(\omega=1, y_{0}=2 \mathrm{ft}\), and \(v_{0}=3 \mathrm{ft} / \mathrm{sec}\), express \(y\) in the form \(A \cos (B t-C)\), and find the amplitude and period of the resulting motion. (b) Determine the times when \(y=0\)-that is, the times when the mass passes through the equilibrium position.

Step-by-Step Solution

Answer

(a) \( y = \sqrt{13} \cos(t - \tan^{-1}(3/2)) \); Amplitude: \( \sqrt{13} \), Period: \( 2\pi \). (b) Times when \( y = 0 \): \( t = \tan^{-1}(3/2) + \frac{\pi}{2} + n\pi \).

Step 1: Write Down Given Functions

We start by writing down the given position function expressed in terms of a sine and cosine function: \ \[ y = y_{0} \cos \omega t + \frac{v_{0}}{\omega} \sin \omega t. \] \ For this problem, the values are \( \omega = 1 \), \( y_{0} = 2 \) ft, and \( v_{0} = 3 \) ft/sec.So the given function becomes: \ \[ y = 2 \cos t + 3 \sin t . \]

Step 2: Convert to Single Trigonometric Function

To express \( y \) in the form \( A \cos(Bt - C) \), we need to convert the sum of sine and cosine functions into a single cosine function. This involves using the identity: \ \[ y = A \cos(Bt - C) = A \cos t \cos C + A \sin t \sin C. \]Comparing terms, we find: \ A \cos C = 2 \ A \sin C = 3 .

Step 3: Find Amplitude and Phase Shift

Solve for \( A \) using: \ \[ A = \sqrt{(A \cos C)^2 + (A \sin C)^2} = \sqrt{2^2 + 3^2} = \sqrt{13}. \]\ Next, solve for \( \cos C \) and \( \sin C \): \[ \cos C = \frac{2}{\sqrt{13}}, \quad \sin C = \frac{3}{\sqrt{13}}. \]This yields \( C \) using \( \tan C = \frac{3}{2} \).

Step 4: Write Function in Desired Form

Now that we have \( A \), \( \cos C \), and \( \sin C \), the function becomes: \ \[ y = \sqrt{13} \cos(t - \tan^{-1} \frac{3}{2}). \] Thus, it's expressed in the form \( A \cos(Bt - C) \) where \( A = \sqrt{13} \), \( B = 1 \), and phase shift is \( C = \tan^{-1} \frac{3}{2} \).

Step 5: Find Period of the Motion

The period \( T \) of the function \( \cos(Bt - C) \) is given by: \ \[ T = \frac{2\pi}{B}. \]Since \( B = 1 \), the period \( T = 2 \pi. \)

Step 6: Determine Times When y = 0

To find when \( y = 0 \), set \( \sqrt{13} \cos(t - \tan^{-1}(3/2)) = 0 \). This implies \( \cos(t - \tan^{-1}(3/2)) = 0 \), giving \ \[ t - \tan^{-1}(3/2) = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}. \] \ Thus, \ \[ t = \tan^{-1}(3/2) + \frac{\pi}{2} + n\pi. \]

Key Concepts

Spring Mass System

Let's start by exploring what a spring mass system is. This physical system involves a mass attached to a spring that can move up and down or back and forth, usually due to external forces like pulling or pushing. What makes it interesting is its ability to exhibit harmonic motion.
### How It Works - **Components**: The system typically consists of three main parts: - A mass object (like a weight) that is free to move. - A spring that the mass is attached to, creating a restoring force when stretched or compressed. - A support or frame to hold the spring steady. - **Restoring Force**: As the mass moves, the spring exerts a restoring force proportional to the displacement from its rest position (Hooke's Law). This force aims to return the mass back to equilibrium. - **Oscillation**: When disturbed, the mass and spring system oscillates around an equilibrium point. It's a classic example of simple harmonic motion, which is a type of periodic motion where the restoring force is directly proportional to the displacement. Understanding the spring mass system is crucial for analyzing the behavior of the mass as it moves, enabling us to predict its position, velocity, and acceleration at any given moment.

Trigonometric Identities

In the context of harmonic motion, trigonometric identities play a key role in transforming expressions. For example, converting the position function of a mass from a sum of sine and cosine terms into a single trigonometric function.- **Sine and Cosine Functions**: These are fundamental in describing oscillatory motion. In the given equation, both functions are used to represent different components of the motion.- **Conversion Identity**: When we have equations like a combination of sine and cosine, we can rewrite them using identities like \[ y = A \cos(Bt - C) = A \cos(t) \cos(C) + A \sin(t) \sin(C) \] This helps to express a more complex oscillating function in terms of simpler trigonometric components.- **Phase Shift**: Using these identities, it's possible to determine the phase shift, which is the horizontal shift of the trigonometric function on a graph, indicative of where the motion starts relative to a standard cycle.With these identities, solving the spring mass system equations becomes more straightforward, allowing us to convert and understand the equations better.

Equilibrium Position

The equilibrium position is a central concept in harmonic motion and particularly in the spring mass system. It refers to the point where the mass experiences no net force from the spring—it's the position at which the spring is neither compressed nor stretched. - **Resting State**: At the equilibrium position, the system's energy is minimal, and this is usually considered the 'neutral' position of the mass. - **Crossing the Equilibrium**: The problem asks when the mass will cross this position, which is significant for understanding the full cycle of motion. When the mass passes through equilibrium, it's momentarily at its maximum speed as it transitions from potential to kinetic energy and vice versa. - **Determining Crossing Times**: Using the equation, we set the displacement to zero and solve for time, thus finding when the mass passes through equilibrium. Understanding the equilibrium position helps to think about how energy is conserved within the system, and how the mass oscillates back and forth.

Amplitude and Period

Amplitude and period are fundamental characteristics of harmonic motion and crucial for describing the behavior of the spring mass system.### Amplitude- **Definition**: It's the maximum displacement from the equilibrium position. In the context of our problem, it determines how far the mass moves from the central point.- **Calculation**: The amplitude in our example is \[ A = \sqrt{13} \] which shows the maximum extent of oscillation from zero.### Period- **Definition**: The period is the time it takes for the mass to complete one full cycle of motion.- **Formula**: It's expressed generally as \[ T = \frac{2\pi}{B} \] Here, with \( B = 1 \), the period simplifies to \[ 2\pi \] meaning the mass repeats its motion every \( 2\pi \) seconds.Knowing the amplitude and period can help predict future behavior of the system, such as its velocity and acceleration at any point in time. It's all about understanding the rhythm of motion.