Problem 10 - Algebra and Trigonometry

Question

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=4 ; \quad T=\frac{\pi}{2} \text { seconds } $$

Step-by-Step Solution

Answer

The displacement function is \(d(t) = 4 \, \cos (4t)\).

Step 1: Identify the harmonic motion equation

The displacement of an object in simple harmonic motion can be described by the function: \[d(t) = a \, \cos \left( \frac{2\pi t}{T} \right)\]Here, \(a\) is the amplitude of motion, \(T\) is the period, and \(t\) is time.

Step 2: Substitute the given values

We are given that \(a = 4\) and \(T = \frac{\pi}{2}\). Substitute these values into the equation: \[d(t) = 4 \, \cos \left( \frac{2\pi t}{\frac{\pi}{2}} \right)\]

Step 3: Simplify the argument of the cosine function

Simplify \( \frac{2\pi t}{\frac{\pi}{2}} \) as follows:\[\frac{2\pi t}{\frac{\pi}{2}} = \frac{2\pi t \times 2}{\pi} = 4t\]Then the displacement function is: \[d(t) = 4 \, \cos (4t)\]

Key Concepts

harmonic motion equation

Simple harmonic motion describes the ideal motion of an object oscillating back and forth regularly about a central point. This motion can be represented mathematically by a harmonic motion equation: Given in the form: d(t) = a \, \cos \left( \frac{2\pi t}{T} \right). Here:

• \(a\) is the amplitude of the motion (the maximum distance from the rest position)
• \(T\) is the period of the motion (the time it takes to complete one cycle)
• \(t\) is the time

The cosine function is used because it provides a smooth, wave-like oscillation pattern, which perfectly models the motion of the object on the spring.

displacement function

To find out how far an object in simple harmonic motion is from its rest position after a given time, we use the displacement function, derived from the harmonic motion equation. In this specific case, we were provided:

• a = 4
• T = \(\frac{\pi}{2}\) seconds

First, substitute these values into the equation: d(t) = 4 \, \cos \left( \frac{2\pi t}{\frac{\pi}{2}} \right). Simplifying this equation simplifies the calculations. For example: The angle inside the cosine function \( \left( \frac{2\pi t}{\frac{\pi}{2}} \right) \) can be simplified. Using the property of fractions in division: \( \frac{2\pi t}{\frac{\pi}{2}} = \frac{2\pi t \times 2}{\pi} = 4t \). Now the displacement function is d(t) = 4 \, \cos(4t). This equation tells us the displacement at any time \(t\) during the simple harmonic motion. It's essential to understand this form because it shows how the position changes over time within one period of the motion. Also, it helps predict future positions or analyze past positions of the object in motion.

trigonometric functions

In simple harmonic motion, trigonometric functions like cosine and sine are crucial. These functions are periodic, meaning they repeat at regular intervals, which makes them perfect for modeling oscillatory motions such as those of a mass-spring system. The cosine function, used in our example, varies between -1 and 1, which helps explain why the displacement varies between -a and a. Some key properties to remember:

• \( \cos (\theta) \) is the horizontal distance or projection on the x-axis in the unit circle
• \( \cos (0) = 1 \) and \( \cos (\pi) = -1 \)
• The cosine function is even, so \( \cos(-\theta) = \cos(\theta) \)

By understanding these properties, it becomes easier to predict and analyze the behavior of objects in simple harmonic motion. For instance, when t is zero (right when the object is released in our example), d(t) calculates using \( \cos(0) \). This is when displacement is its maximum positive value, which is just \( a \). Moreover, the patterns in these functions' behavior over time can be used to predict future motion and assess cyclical behaviors in physical applications.