# Maximum speed and magnitude of the acceleration of a spring?

• Howard Fox
In summary, the maximum speed and acceleration of a guitar string executing simple harmonic motion can be found by setting the equation for velocity (dx/dt) and acceleration (dv/dt) equal to zero and one, respectively, as the maximum values for these quantities occur when the cosine function is at its maximum value of one.
Howard Fox

## 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?

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.

Howard Fox

## What is the maximum speed of a spring?

The maximum speed of a spring is determined by the amplitude of the oscillations and the mass of the object attached to the spring. It can be calculated using the equation v = ωA, where v is the maximum speed, ω is the angular frequency, and A is the amplitude.

## What is the magnitude of acceleration of a spring?

The magnitude of acceleration of a spring can be calculated using the equation a = -ω²x, where a is the acceleration, ω is the angular frequency, and x is the displacement from equilibrium. The acceleration is directly proportional to the square of the angular frequency and inversely proportional to the displacement.

## How does the mass of an object affect the maximum speed and acceleration of a spring?

The mass of an object attached to a spring affects both the maximum speed and acceleration. A heavier object will result in a lower maximum speed and a higher magnitude of acceleration, while a lighter object will result in a higher maximum speed and a lower magnitude of acceleration.

## What factors can affect the maximum speed and acceleration of a spring?

Besides the amplitude and mass, other factors that can affect the maximum speed and acceleration of a spring include the stiffness of the spring, damping forces, and external forces acting on the object attached to the spring.

## How can the maximum speed and acceleration of a spring be measured?

The maximum speed and acceleration of a spring can be measured using various instruments such as a stopwatch, a motion sensor, or a high-speed camera. These measurements can then be used to calculate the values using the appropriate equations.

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