A weight attached to a spring is at its lowest point, 9 inches below equilibrium, at time [tex]$t=0$[/tex] seconds. When the weight is released, it oscillates and returns to its original position at [tex]$t=3$[/tex] seconds. Which of the following equations models the distance, [tex]$d$[/tex], of the weight from its equilibrium after [tex]$t$[/tex] seconds?
A. [tex]$d=-9 \cos \left(\frac{\pi}{3} t\right)$[/tex]
B. [tex]$d=-9 \cos \left(\frac{2 \pi}{3} t\right)$[/tex]
C. [tex]$d=-3 \cos \left(\frac{\pi}{9} t\right)$[/tex]
D. [tex]$d=-3 \cos \left(\frac{2 \pi}{9} t\right)$[/tex]
Answer (2)
The distance is modeled by a cosine function: d = A \tims \tcos ( Bt ) .
The amplitude is A = − 9 inches.
The period is T = 6 seconds, so B = 3 π .
The equation is d = − 9 \tims \tcos ( 3 π t ) .
The final answer is d = − 9 \tcos ( 3 π t ) .
Explanation
Understanding the Problem We are given that a weight attached to a spring is at its lowest point, 9 inches below equilibrium, at time t = 0 seconds. When the weight is released, it oscillates and returns to its original position at t = 3 seconds. We need to find the equation that models the distance, d , of the weight from its equilibrium after t seconds.
Setting up the Equation Since the weight is at its lowest point at t = 0 , we can model the distance using a cosine function of the form d = A cos ( Bt ) , where A is the amplitude and B is related to the period.
Finding the Amplitude The amplitude A is the distance from the equilibrium position to the lowest point, which is − 9 inches (since it's below equilibrium). So, A = − 9 .
Determining the Period The weight returns to its original position at t = 3 seconds. This means it completes half of its oscillation in 3 seconds. Therefore, the full period of oscillation is T = 2 × 3 = 6 seconds.
Calculating B The period T is related to B by the formula T = B 2 π . We have T = 6 , so 6 = B 2 π . Solving for B , we get B = 6 2 π = 3 π .
Final Equation Now we substitute A = − 9 and B = 3 π into the equation d = A cos ( Bt ) to get d = − 9 cos ( 3 π t ) .
Matching the Answer Comparing our equation with the given options, we see that it matches the first option: d = − 9 cos ( 3 π t ) .
Examples
Understanding oscillatory motion is crucial in various fields, such as physics and engineering. For example, designing suspension systems in cars involves modeling the oscillations of springs to ensure a smooth ride. Similarly, analyzing the vibrations of structures like bridges requires understanding oscillatory motion to prevent resonance and potential collapse. The equation we derived helps predict the position of an object undergoing simple harmonic motion, which is a fundamental concept in these applications.
The correct equation modeling the distance d of the weight from its equilibrium after t seconds is d = − 9 cos ( 3 π t ) . This derives from understanding the conditions of amplitude and period in simple harmonic motion. Therefore, the answer is option A.
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