Question

1. the acceleration of the oscillator is a(t)=v(t). find and graph the acceleration function.

a. t = 0, 2π, 4π,... ; the position is (0, - 36), (2π, - 36), (4π, - 36),...
b. t=\frac{3pi}{2},\frac{5pi}{2},\frac{7pi}{2},... ; the position is (\frac{3pi}{2}, - 72),(\frac{5pi}{2},0),(\frac{7pi}{2}, - 72),...
c. t=pi,3pi,5pi,... ; the position is (pi,0),(3pi,0),(5pi,0),...
d. t=\frac{pi}{2},\frac{3pi}{2},\frac{5pi}{2},... ; the position is (\frac{pi}{2},0),(\frac{3pi}{2}, - 72),(\frac{5pi}{2},0),...

Explanation:

Step1: Analyze the problem

The problem is about finding and graphing the acceleration - function of an oscillator. We need to use the given information about positions at different times to understand the nature of the acceleration function. However, the problem seems to be incomplete as the acceleration function \(a(t)=v^{\prime}(t)\) is given but no information about the velocity - function \(v(t)\) or the position - function \(s(t)\) is sufficient to actually calculate values for graphing. But we can analyze the given time - position pairs.

Step2: Check the options

We are given time values and corresponding position values in the options. We know that in simple harmonic motion, the acceleration \(a =-\omega^{2}x\) (where \(x\) is the displacement from the equilibrium position and \(\omega\) is the angular frequency). When the position \(x = 0\), the acceleration \(a = 0\), and when the position is at its maximum or minimum, the acceleration is at its maximum in magnitude.
Looking at the options, we need to find a pattern that relates to the behavior of an oscillator's acceleration. In option A, at \(t = 0,2\pi,4\pi,\cdots\), the position has non - zero values \((0, - 36),(2\pi,-36),(4\pi,-36)\cdots\). In option B, we have a pattern of positions at different times. In option C, at \(t=\pi,3\pi,5\pi,\cdots\), the position is \((\pi,0),(3\pi,0),(5\pi,0)\cdots\) which means at these times the acceleration is \(0\) (since \(x = 0\)). In option D, we have a set of time - position pairs.
Since the problem is asking for the times when we can analyze the acceleration based on the position, and we know that acceleration is \(0\) when the position is at the equilibrium (\(x = 0\)). The times \(t=\pi,3\pi,5\pi,\cdots\) are the times when the position is at the equilibrium position \((x = 0)\) and thus the acceleration is \(0\) at these times.

Answer:

C. \(t=\pi,3\pi,5\pi,\cdots\); the position is \((\pi,0),(3\pi,0),(5\pi,0)\cdots\)