Maximum speed and magnitude of the acceleration of a spring?

AI Thread Summary
The discussion focuses on calculating the maximum speed and acceleration of a guitar string executing simple harmonic motion, described by the equation x(t)=Asin(wt+φ). The angular frequency is given as ω = 2.76 × 10^3 s−1 and the amplitude A is 1.60 mm. The maximum speed occurs when the displacement x is zero, while the maximum acceleration occurs at the maximum displacement A. The key equations for velocity and acceleration are provided, with clarification that the maximum value of the cosine function is one. The conversation emphasizes understanding the behavior of the trigonometric functions involved in the motion.
Howard Fox
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Homework Statement


If a midpoint of a guitar string executes simple harmonic motion following the form x(t)=Asin(wt+φ), and its angular frequency is ω = 2.76 × 103 s−1 and A=1.60mm. What is then its maximum speed of the string during this motion? And what is the maximum magnitude of the acceleration of the spring?

Homework Equations


x(t)=Asin(wt+φ)
dx/dt = Aω cos(ωt + φ)
dv/dt = −Aω2 sin(ωt + φ)

The Attempt at a Solution


I know that the maximum velocity should happen when x=0 and the maximum acceleration is when displacement is greatest, but I am not sure on how to proceed. Should I set the second equation I listed above equal to zero to find the velocity and the third equation equal to 1.60mm to find the acceleration?
Thank you for your help.
 
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Howard Fox said:
Should I set the second equation I listed above equal to zero ...
No, just look at it. The right side is maximum when the cosine is maximum. What is the maximum value of a cosine, any cosine?
 
kuruman said:
No, just look at it. The right side is maximum when the cosine is maximum. What is the maximum value of a cosine, any cosine?
One?
 
What is dx/dt ?
 
Howard Fox said:
One?
One.
 
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