13.11. EXERCISES

Which of the following examples represent periodic motion?\n(a) A swimmer completing one (return) trip from one bank of a river to the other and back.\n(b) A freely suspended bar magnet displaced from its N-S direction and released.\n(c) A hydrogen molecule rotating about its center of mass.\n(d) An arrow released from a bow.

In this exercise, students are asked to identify examples of periodic motion. Periodic motion is characterized by a movement that repeats itself at regular intervals. For instance, a swimmer's round trip in a river (option a) is periodic, as it involves a continuous back-and-forth pattern. A freely suspended bar magnet (option b) also demonstrates periodic behavior when returned to its resting position after being displaced. In contrast, an arrow released from a bow (option d) does not exhibit periodic motion; it moves in one direction without returning.

Think of a swing in a playground. When someone pushes the swing, it goes back and forth continuously. Just like that, the swimmer (option a) completes a repetitive cycle, returning to the same spot, which makes it periodic.

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?\n(a) the rotation of earth about its axis.\n(b) motion of an oscillating mercury column in a U-tube.\n(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.\n(d) general vibrations of a polyatomic molecule about its equilibrium position.

In this exercise, students differentiate between nearly simple harmonic motions and those that are periodic but not simple harmonic. Simple harmonic motion (SHM) occurs when a restoring force is directly proportional to the displacement and acts towards the equilibrium position. For instance, the motion of an oscillating mercury column in a U-tube (option b) and the ball bearing in a bowl (option c) exhibit SHM characteristics. Meanwhile, the Earth's rotation (option a) is periodic but not SHM since it does not involve restorative forces acting towards an equilibrium position.

Imagine a mass on a spring. When you pull it and let go, it oscillates back and forth quickly around its original position before coming to rest. This is similar to the oscillating mercury column (option b) where the liquid moves up and down following a harmonic pattern.

Fig. 13.18 depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

In this exercise, students examine graphical representations of linear motion (x-t plots) to identify periodic behavior. A periodic motion will show a repeating pattern over time in the x-t graph. For instance, sinusoidal waves represent simple harmonic motion and exhibit periodicity. By analyzing the plots, students should determine which ones depict consistent repetitions, and calculate the period of those motions.

Consider the motion of a swing again, moving back and forth. If you were to plot its position over time, it would show a predictable pattern, allowing one to easily identify the swing's periods of motion\u2014much like looking at the plots in this exercise.

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (\u03c9 is any positive constant):\n(a) sin \u03c9t \u2013 cos \u03c9t\n(b) sin^3 \u03c9t\n(c) 3 cos (\u03c0/4 \u2013 2\u03c9t)\n(d) cos \u03c9t + cos 3\u03c9t + cos 5\u03c9t\n(e) exp(\u2013\u03c9\u00b2t\u00b2)\n(f) 1 + \u03c9t + \u03c9\u00b2t\u00b2.

This exercise encourages students to categorize different mathematical functions based on their characteristics. Functions that yield sinusoidal outputs, like (a) and (b), generally indicate simple harmonic motion. Option (c) and (d) might be periodic functions but differ from simple harmonic due to their composite nature. Non-periodic functions, such as (e) and (f), do not show repeating behavior. Students will need to analyze each function mathematically to determine its periodicity and categorize them accordingly.

Think of a song composed of various notes. Notes played in harmony (like option (a) and (b)) will create a beautiful melody, much like SHM. However, an irregular sound might occur in the background (like option (e)) disrupting any periodic flow, much like a non-periodic function.

A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration, and force on the particle when it is\n(a) at the end A,\n(b) at the end B,\n(c) at the mid-point of AB going towards A,\n(d) at 2 cm away from B going towards A,\n(e) at 3 cm away from A going towards B, and\n(f) at 4 cm away from B going towards A.

In this exercise, students analyze a simple harmonic motion scenario by applying their knowledge of the signs for velocity, acceleration, and force throughout the motion. At endpoints A and B, the velocity will be zero as the particle changes direction, but acceleration and force will be directed towards the equilibrium position. As the particle moves away from the equilibrium position, the force and acceleration will indicate the direction of movement, reinforcing that they always direct back towards the center.

Imagine pulling a rubber band and letting it go. At the moment before you let go (at point A), the band isn't moving (velocity is zero), but it's all stretched and wants to snap back (acceleration and force), which is similar to a particle's behavior at points A and B.

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