Starting at its rightmost position, it takes 2 seconds for the pendulum of a grandfather clock to swing a horizontal distance of 18 inches from right to left and 2 seconds for the pendulum to swing back from left to right. Which of the following equations models $d$, the horizontal distance in inches of the pendulum from the center as a function of time, $t$, in seconds? Assume that right of center is a positive distance and left of center is a negative distance.
A. [tex]d=9 \cos \left(\frac{\pi}{4} t\right)[/tex]
B. [tex]d=9 \cos \left(\frac{\pi}{2} t\right)[/tex]
C. [tex]d=18 \cos \left(\frac{\pi}{4} t\right)[/tex]
D. [tex]d=18 \cos \left(\frac{\pi}{2} t\right)[/tex]
Answer (1)
The pendulum's amplitude is 9 inches and its period is 4 seconds.
The general form of the equation is d = A cos ( Bt ) , where A is the amplitude and the period is B 2 π .
Calculate B using the period: B = 4 2 π = 2 π .
The equation that models the pendulum's motion is d = 9 cos ( 2 π t ) .
Explanation
Understanding the Problem We are given that the pendulum swings 18 inches from right to left in 2 seconds and back to the right in another 2 seconds. This means the total time for one complete swing (right to left and back) is 4 seconds. We also know that the amplitude of the swing is half the total distance, which is 18/2 = 9 inches. Since the pendulum starts at its rightmost position, we can model the distance d as a cosine function of time t .
Setting up the Equation The general form of the equation is d = A cos ( Bt ) , where A is the amplitude and the period is B 2 π . In this case, the amplitude A is 9 inches, and the period is 4 seconds.
Solving for B and Final Equation We have the period 4 = B 2 π , so we can solve for B : B = 4 2 π = 2 π Therefore, the equation is d = 9 cos ( 2 π t ) .
Final Answer Comparing our equation d = 9 cos ( 2 π t ) with the given options, we see that it matches the second option.
Examples
Pendulums are used in various applications, such as metronomes for music, seismometers for detecting earthquakes, and even in some amusement park rides. Understanding the physics of pendulum motion, including the relationship between its length, period, and amplitude, allows engineers to design and optimize these systems. For example, knowing the period of a pendulum is crucial for accurately timing musical beats in a metronome or for ensuring the stability of a pendulum-based clock. The equation we derived helps predict the pendulum's position at any given time, which is essential for these applications.
