A spider sits on its web, undergoing simple harmonic motion with amplitude A.What fraction of each cycle does the spider spend at positions with x>0.75A? Hint: Use the analog of uniform circular motion.
To determine the fraction of each cycle that the spider spends at positions with \(x > 0.75A\), we can analyze the motion of a spider using the analog of uniform circular motion.
Simple harmonic motion (SHM) is often compared to the motion of a point on the edge of a circle rotating at a uniform speed. This analogy helps us relate the position of the spider on its web to angles on a circle.
Here's the step-by-step approach to finding the fraction of each cycle the spider spends at positions with \(x > 0.75A\):
1. Start by imagining the spider's web as a circular path with radius \(A\).
2. Consider the spider at its maximum displacement, \(x = A\), as being at the point of the circle directly to the right of the center. This is our reference position.
3. Any position of the spider can be represented by an angle relative to this reference position. By using trigonometry, we can relate these angles to the position of the spider.
4. Apply the analogy of uniform circular motion to the spider's motion. Since SHM is symmetrical, the spider's journey can be divided into four equal parts: going from \(x = 0\) to \(x = A\), returning to \(x = 0\), going from \(x = 0\) to \(x = -A\), and returning to \(x = 0\) again.
5. Focus on the part of the motion where \(x > 0.75A\) which corresponds to \(x > 0.75\) times its maximum displacement.
6. Calculate the corresponding angle on the circle for \(x = 0.75A\) using trigonometry. Let this angle be \(\theta\).
7. Determine the fraction of the circular path that corresponds to \(x > 0.75A\) by finding the ratio of the angle \(\theta\) to a full circle (360 degrees or \(2\pi\) radians).
8. The fraction obtained in step 7 represents the fraction of each cycle the spider spends at positions with \(x > 0.75A\).
Following these steps will help you determine the fraction of each cycle the spider spends at positions with \(x > 0.75A\) in a simple harmonic motion.