Problem 57 - Contemporary Precalculus

Question

A pendulum swings uniformly back and forth, taking 2 seconds to move from the position directly above point \(A\) to the position directly above point \(B\). (Check your book to see image) The distance from \(A\) to \(B\) is 20 centimeters. Let \(d(t)\) be the horizontal distance from the pendulum to the (dashed) center line at time \(t\) seconds (with distances to the right of the line measured by positive numbers and distances to the left by negative ones). Assume that the pendulum is on the center line at time \(t=0\) and moving to the right. Assume that the motion of the pendulum is simple harmonic motion. Find the rule of the function \(d(t)\)

Step-by-Step Solution

Answer

Answer: The function is \(d(t) = 10 \cos(\frac{\pi}{2}t + \frac{\pi}{2})\).

Step 1: Determine the amplitude of the motion

Since the pendulum has a simple harmonic motion, the maximal displacement from the center point (called amplitude) is half the distance between points A and B, i.e., \(\frac{1}{2} \times 20 \;\text{cm}\). Thus, the amplitude is \(10 \;\text{cm}\).

Step 2: Determine the period of the motion

According to the problem, the pendulum takes 2 seconds to move from point A to point B, which is half the cycle of the motion. Therefore, the period of the motion, i.e., the time needed for the pendulum to complete one full cycle, is \(2 \times 2 = 4 \;\text{seconds}\).

Step 3: Write a function

Since the pendulum starts at the center line and moves to the right, we need a function that starts at 0 and has a value of 10 at \(t=1\). A cosine function is suitable for this task, as it starts at maximum value when the argument is 0. So, the function has the form: $$ d(t) = A \cos(\frac{2\pi}{T}t + p) $$ Where \(A\) is the amplitude, \(T\) is the period, and \(p\) is the phase shift. Now we can plug in the values for amplitude and period that we determined in steps 1 and 2: $$ d(t) = 10 \cos(\frac{2\pi}{4}t + p) $$ Since the pendulum starts at the center line, we know that \(d(0) = 0\). Plugging in \(t=0\), we get: $$ 0 = 10 \cos(p) $$ This means that \(p\) must be \(\pm \frac{\pi}{2}\), or in general, \(p = \pm \frac{\pi}{2} + 2n\pi\), where \(n\) is an integer. Since the pendulum is moving to the right from the center line, we can choose the positive phase shift: $$ p = \frac{\pi}{2} $$

Step 4: Construct the final function

Plugging the phase shift into the function, we get: $$ d(t) = 10 \cos(\frac{2\pi}{4}t + \frac{\pi}{2}) $$ Simplifying, we get the final rule of the function \(d(t)\): $$ d(t) = 10 \cos(\frac{\pi}{2}t + \frac{\pi}{2}) $$ This function describes the horizontal distance from the pendulum to the center line at time \(t\) seconds.

Key Concepts

Pendulum Motion

Pendulums are fascinating examples of periodic motion, moving back and forth in a regular path. This type of repetitive movement is known as 'simple harmonic motion.' Understanding simple harmonic motion involves observing how an object like a pendulum swings through its path in equal intervals of time.
When a pendulum swings, it travels from its highest point on one side (let's call it point A) to the highest point on the other (point B). During this swing, it moves through an equilibrium position right in the middle. This is the center line, and as the pendulum crosses this line, it's where it has the most kinetic energy and the least potential energy.

• In simple harmonic motion, every complete swing from point A to point B and back is called one cycle.
• The time it takes to complete a half cycle (from A to B or back) can help us determine the full period of the pendulum's swinging motion.

Recognizing pendulum motion as simple harmonic simplifies the mathematical modeling. This allows us to predict its behavior using specific equations and helps us understand similar motions in different systems.

Cosine Function

In mathematics, the cosine function is a crucial tool for describing periodic phenomena like pendulum motion. The cosine function is a trigonometric function, often paired with sine, that can describe oscillations due to its repetitive wave-like properties.
The function is formally expressed as a function of an angle and is periodic, meaning it repeats its values in regular intervals (known as the period). When modeling the pendulum's motion, the cosine function is particularly useful because it starts at its peak value when the angle is zero.

• The general form of a cosine function used in modeling is: \(d(t) = A \cos(\omega t + p)\)where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(p\) is the phase shift.
• For our pendulum, since it starts at the equilibrium line at \(t=0\), the function's addition of a phase shift is necessary to reflect this starting position accurately.

Accuracy in using the cosine function helps give a clear description of how the pendulum moves over time, providing insights into other oscillatory behaviors, both simple and complex.

Amplitude and Period

In any oscillating system, understanding amplitude and period is fundamental. These two parameters help shape the motion's description, making it possible to calculate and predict the movement precisely.

• Amplitude: This is the maximum distance from the equilibrium (center line) that the pendulum reaches during its swing. In our pendulum example, the amplitude is half the distance between points A and B, calculated to be 10 centimeters. This tells us the farthest the pendulum swings from the center.
• Period: The period is the time it takes for the pendulum to complete a full cycle, returning to its starting position. Given our pendulum's description, it takes two seconds to go from A to B, so the full period is double that, making it four seconds for a complete cycle.

These measurements are not just numbers. They provide essential insights into the dynamics of the pendulum's motion. The amplitude gives us a sense of the pendulum's energy, while the period lets us predict how long the motion will take, an important part of designing systems that rely on rhythmic, predictable movement.