Problem 55 - Contemporary Precalculus

Question

Weight is pulled 6 centimeters above equilibrium, and the initial movement (at \(t=0\) ) is downward. [Hint: Think cosine. \(]\)

Step-by-Step Solution

Answer

Question: Write the equation that describes the position of the weight at any time t when it is pulled 6 centimeters above its equilibrium position and released. Answer: The equation for the position of the weight at time t is \(x(t) = 6 \cos(\omega t)\), where \(\omega\) is the angular frequency.

Step 1: Finding the amplitude

The weight is pulled 6 centimeters above the equilibrium, which means the maximum displacement from the equilibrium position is 6 cm. Therefore, the amplitude \(A\) is 6 cm

Step 2: Determining the angular frequency \(\omega\)

Since we are not given any information about the frequency or the period of the motion, we cannot determine the exact value of \(\omega\). However, we can still leave it as a variable in our equation.

Step 3: Finding the phase shift \(\phi\)

At time \(t=0\), the weight is at its maximum displacement (6 cm above the equilibrium) and starts moving downward. In a standard cosine function, it starts at the maximum value and moves downward. Thus, there is no phase shift. We can set \(\phi\) to 0.

Step 4: Writing the equation

Now we have all the necessary information to write the position function \(x(t)\) for the weight's motion: $$x(t)=A \cos(\omega t + \phi) = 6 \cos(\omega t)$$ So, the equation describing the position of the weight at time \(t\) is \(x(t)=6 \cos(\omega t)\), where \(\omega\) is the angular frequency.

Key Concepts

Understanding Amplitude

Amplitude in simple harmonic motion is the maximum distance that an object moves from its equilibrium position. In this context, imagine the starting point of a weight hanging from a spring or a pendulum at rest. When you pull it slightly away and then release it, the distance it moves away and back to the rest position is the amplitude.

• The amplitude gives you an idea of how far the object moves during oscillation.
• It is always a positive number, regardless if the motion is upwards or downwards.

In our exercise, the weight is pulled 6 centimeters above the equilibrium position, making the amplitude 6 cm. This tells us that the weight moves 6 cm from the center point of its rest position when it’s set into motion.

Grasping Angular Frequency

Angular Frequency, denoted by the symbol \(\omega\), is a measure of how quickly an object oscillates in simple harmonic motion. Think about it as the speed of the oscillation, but it’s measured in radians per second instead of the regular speed units you might know.

• Angular frequency connects the frequency of oscillations with the periodic motion.
• While not provided in this problem, typically, \(\omega = 2\pi f\), where \(f\) is the frequency.

In cases where no exact frequency or period is given, like in our exercise, \(\omega\) can be left as a variable in the equation for simplicity. It plays an important role in fully defining the behavior of oscillation in mathematical terms.

Decoding Phase Shift

The phase shift \(\phi\) refers to the horizontal shift in the graph of the cosine function. It tells us how much the graph is shifted away from its usual starting position, which is typically at the peak (maximum).

• A phase shift occurs if the start of the motion doesn't coincide with the standard starting point of a cosine wave.
• If the motion at \(t=0\) starts exactly at a peak, we typically set \(\phi = 0\).

In the exercise, since the weight is at the maximum displacement and follows the downward path immediately, there is no need for a phase shift. The angle starts from the standard cosine graph peak, meaning \(\phi = 0\). This simplifies our equation without altering the motion description.

Applying the Cosine Function

The cosine function is well-suited for modeling simple harmonic motion, especially when the initial movement begins from a maximum displacement. It may not be the first trigonometric function you link to oscillation, but it provides a natural start from a peak in many situations.

• The general form of the cosine function used in harmonic motion is \(x(t) = A \cos(\omega t + \phi)\).
• This equation helps predict the position \(x(t)\) at any given time \(t\).

For the exercise at hand, using \(x(t) = 6 \cos(\omega t)\) effectively describes how the weight moves over time. Even though \(\omega\) is unknown, this equation communicates how the weight returns to its equilibrium repeatedly in a simple pattern characterized by amplitude.