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Chapter 13: Problem 7

When in its cycle is the acceleration of an undamped simple harmonic oscillator zero? When is the velocity zero?

Short Answer

Expert verified
The acceleration of an undamped simple harmonic oscillator is zero when the system is at its equilibrium position, i.e., in the middle of its cycle. The velocity is zero at the extreme points of the cycle, where the object changes direction.

Step by step solution

01

Define the properties of Simple Harmonic Oscillators

In a Simple Harmonic Oscillator (SHO), the acceleration and the velocity are periodic functions. The acceleration \(a\) is given by \(a= -\omega^2 * x\), where \(x\) is the displacement from the equilibrium position and \(\omega\) is the angular frequency. Likewise, the velocity \(v\) is given by \(v = \omega * sqrt(\(A^2 - x^2)\), where \(A\) is the amplitude of the motion.
02

Calculate when the acceleration equals zero

The acceleration equals zero when the displacement \(x\) equals zero because of the multiplication in the acceleration formula. This happens twice in every cycle of the SHO, when the system passes through its equilibrium position.
03

Calculate when the velocity equals zero

The velocity equals zero when the term under the square root in the velocity equation equals zero, which occurs when the displacement \(x\) equals the amplitude \(A\). This means that the velocity is zero at the extreme points of the SHO's cycle as the oscillator changes its direction of motion.

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