Select the correct answer.
Jackson is conducting an experiment for his Physics class. He attaches a weight to the bottom of a metal spring. He then pulls the weight down so that it is a distance of 6 inches from its equilibrium position. Jackson then releases the weight and finds that it takes 4 seconds for the spring to complete one oscillation.
Which function best models the position of the weight?
A. [tex]s(t)=-6 \cos (2 \pi t)[/tex]
B. [tex]s(t)=-6 \cos \left(\frac{\psi}{2} t\right)[/tex]
C. [tex]s(t)=6 \sin \left(\frac{\pi}{2} t\right)[/tex]
D. [tex]x(t)=6 \sin (2 \pi t)[/tex]
Answer (2)
The amplitude of the oscillation is 6 inches.
The angular frequency is calculated as ω = T 2 π = 2 π .
Since the weight is released from its maximum displacement, use a negative cosine function: s ( t ) = − A cos ( ω t ) .
Substitute the values of A and ω to get the function: s ( t ) = − 6 cos ( 2 π t ) .
The correct answer is s ( t ) = − 6 cos ( 2 π t ) .
Explanation
Understanding the Problem We are given that a weight attached to a spring is pulled down 6 inches from its equilibrium position and then released. The spring completes one oscillation in 4 seconds. We need to find the function that models the position of the weight.
Finding Amplitude and Period The amplitude of the oscillation is the initial displacement, which is 6 inches. The period of the oscillation is 4 seconds.
Choosing the Correct Function The general form of the function that models the position of the weight is given by s ( t ) = A cos ( ω t + ϕ ) where A is the amplitude, ω is the angular frequency, and ϕ is the phase shift. Since the weight is released from its maximum displacement, a cosine function is appropriate.
Calculating Angular Frequency The angular frequency ω is related to the period T by the formula ω = T 2 π Since T = 4 seconds, we have ω = 4 2 π = 2 π
Determining the Function Since the weight is initially pulled down 6 inches, the initial position is -6. Therefore, we should use a negative cosine function: s ( t ) = − A cos ( ω t ) Substituting the values of A and ω into the equation, we get s ( t ) = − 6 cos ( 2 π t )
Selecting the Correct Answer Comparing the derived function with the given options, we see that option B matches our result: s ( t ) = − 6 cos ( 2 π t ) Therefore, the correct answer is B.
Examples
Spring oscillations are a fundamental concept in physics and engineering. They appear in various real-world scenarios, such as the suspension systems in cars, the motion of a pendulum in a clock, and the vibration of atoms in a solid. Understanding how to model these oscillations with mathematical functions allows engineers to design systems that can withstand vibrations, such as bridges and buildings, or to create devices that rely on precise oscillatory motion, such as musical instruments and electronic oscillators. By analyzing the amplitude, period, and phase shift of an oscillation, we can predict the behavior of a system and optimize its performance.
The function that models the position of the weight on the spring is s ( t ) = − 6 cos ( 2 π t ) . This corresponds to option B, where the amplitude is 6 inches and the angular frequency is \frac{\pi}{2}. This choice correctly represents the weight's motion after being pulled down and released.
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