Problem 21 - Calculus Early Transcendentals

Question

A mass hanging from a spring is set in motion, and its ensuing velocity is given by \(v(t)=2 \pi \cos \pi t\) for \(t \geq 0 .\) Assume that the positive direction is upward and that \(s(0)=0\) a. Determine the position function, for \(t \geq 0\) b. Graph the position function on the interval [0,4] c. At what times does the mass reach its low point the first three times? d. At what times does the mass reach its high point the first three times?

Step-by-Step Solution

Answer

In summary, the mass reaches its low points the first three times at \(t = 1, 3,\) and \(5\), and it reaches its high points the first three times at \(t = 2, 4,\) and \(6\). The position function is given by \(s(t) = 2\sin(\pi t)\).

Step 1: Determine the position function, for \(t \geq 0\)

To find the position function \(s(t)\), we integrate the velocity function \(v(t)\): \(s(t) = \int v(t) dt = \int 2\pi \cos \pi t dt\) Now, integrate using substitution. Let \(u = \pi t\), so \(du/dt = \pi\) and \(dt = \frac{1}{\pi} du\). Then, \(s(t) = \int 2\pi \cos u \frac{1}{\pi} du = 2 \int \cos u du\) Next, integrate the cosine function: \(s(t) = 2 \sin u + C = 2 \sin(\pi t) + C\) Apply the initial condition \(s(0) = 0\): \(0 = 2 \sin(\pi(0)) + C \Rightarrow C=0\). Thus, the position function is: \(s(t) = 2\sin(\pi t)\).

Step 2: Graph the position function on the interval \([0,4]\)

Utilize graphing software or a graphing calculator to plot the position function \(s(t) = 2\sin(\pi t)\) on the interval \([0, 4]\). Remember that this is a sinusoidal function with amplitude \(2\) and period \(2\) (since the argument of the sine function is \(\pi t\)).

Step 3: Find when the mass reaches its low point the first three times

To find when the mass reaches its low point, we need to find when the position function has a local minimum. Since \(s(t) = 2\sin(\pi t)\), a local minimum occurs when the sine function is at its minimum, which is \(-1\). Set the position function equal to \(-2\): \(2\sin(\pi t) = -2 \Longrightarrow \sin(\pi t) = -1\) This occurs when \(\pi t = (2n+1)\pi\) for \(n=0,1,2,...\) Solve for \(t\): \(t = \frac{(2n+1)\pi}{\pi} = 2n+1\) So the low points occur at \(t=1, 3, 5,...\). Since we only need the first three, the mass reaches its low point at \(t=1, 3,\) and \(5\).

Step 4: Find when the mass reaches its high point the first three times

Similar to step 3, we find when the mass reaches its high point by finding when the position function has a local maximum. Since \(s(t) = 2\sin(\pi t)\), a local maximum occurs when the sine function is at its maximum value, which is \(1\). Set the position function equal to \(2\): \(2\sin(\pi t) = 2 \Longrightarrow \sin(\pi t) = 1\) This occurs when \(\pi t = n\pi\) for \(n=0,1,2,...\) and \(n \neq 0\) since it's the maximum value. Solve for \(t\): \(t = \frac{n\pi}{\pi} = n\) So the high points occur at \(t=2, 4, 6,...\). Since we only need the first three, the mass reaches its high point at \(t=2, 4,\) and \(6\).

Key Concepts

Position Function

The position function is a key concept in understanding motion. It describes how the position of an object changes over time. In this case, for a mass on a spring, the position function is found by integrating the given velocity function. The velocity function is related to the rate of change of the position. Since the velocity function is given as \( v(t) = 2\pi \cos(\pi t) \), integrating this function helps to find the position function \( s(t) \).

The integration involves:

• Substitution: Using \( u = \pi t \), we find \( du/dt = \pi \), and thus \( dt = \frac{1}{\pi} du \).
• Integral calculation: \( s(t) = \int 2\pi \cos(\pi t) dt = 2 \int \cos(u) du = 2\sin(u) + C \).
• Applying initial condition: Given \( s(0) = 0 \), solve for \( C \), leading to \( C = 0 \).

This results in the position function \( s(t) = 2\sin(\pi t) \). Integrating the velocity function reveals important behavior about how the mass moves along the spring over time.

Velocity Function

Velocity describes how quickly an object's position changes with time. In this exercise, the velocity function is given as \( v(t) = 2\pi \cos(\pi t) \). The function provides essential insights into the motion, particularly:

• The cosine function indicates the velocity varies periodically due to the spring's oscillatory nature.
• The amplitude of \( 2\pi \) means that the peak speed (positive or negative) is \( 2\pi \) units.
• The frequency is determined by the \( \pi t \), indicating each cycle completes as \( t \) changes by 2 units.
• A negative velocity means motion in the opposite direction (here, downward), while positive means upward.

Velocity is integral in determining other aspects of motion, such as identifying when the mass changes direction or when it reaches certain points during its motion.

Sine Function

The sine function plays a pivotal role in sinusoidal motion, which is often observed in systems like springs. The position function derived as \( s(t) = 2\sin(\pi t) \) signifies that the mass's position oscillates following a sine curve. Here are essential characteristics:

• The sine function has default range \(-1\) to \(1\), leading to position oscillating from \(-2\) to \(2\).
• Its amplitude is \(2\), meaning the mass's maximum displacement is 2 units from the equilibrium.
• The period of the sine function, influenced by \( \pi t \), dictates a full cycle every \(2\) time units.
• The sine’s periodic nature means consistent oscillations over time, mirroring the predictable rhythmic motion of springs.

Understanding the sine function is crucial in visualizing the rhythmic, harmonic motion in such physical systems.

Periodic Motion

Periodic motion is a type of motion that repeats itself at regular intervals. This concept is a cornerstone in analyzing systems like springs and pendulums. In this exercise, periodic motion is demonstrated in the way the mass on the spring oscillates.

The characteristics include:

• Repetition: The motion repeats every \(2\) units of time, aligning with the period of the sine function in \( s(t) = 2\sin(\pi t) \).
• Oscillation: The mass moves back and forth about an equilibrium position, resulting in alternate high and low points.
• Predictable Patterns: By knowing the period and amplitude, one can predict future positions and velocities of the mass.

Periodic motion is integral for understanding how restoring forces like those from a spring act to repeatedly return systems to a stable state. This helps to predict system behavior over time in engineering and natural systems.