CBSE Class 11 Physics Chapter 14 Oscillations NCERT Solutions

The examples that represent periodic motion are (b) and (c).

Let's analyze each option:

1. A swimmer completing one (return) trip from one bank of a river to the other and back: This motion is not necessarily periodic. While the swimmer moves back and forth, the time taken for the trip to the other bank might not be the same as the time taken for the return trip. For a motion to be periodic, it must repeat itself exactly in equal intervals of time.
2. A freely suspended bar magnet displaced from its N-S direction and released: This motion is periodic. When displaced and released, the magnet will oscillate about its equilibrium position (N-S direction) with a definite time period, repeating its motion.
3. A hydrogen molecule rotating about its center of mass: This motion is periodic. As the molecule rotates, it returns to its original position and orientation after a fixed interval of time, thus exhibiting periodic motion.
4. An arrow released from a bow: This motion is not periodic. The arrow moves in a single direction (forward) and does not return to its starting point to repeat the motion.

Therefore, options (b) and (c) represent periodic motion.

Let's classify each motion:

1. The rotation of Earth about its axis: This is a periodic motion, but not simple harmonic motion (SHM). The Earth completes one rotation in approximately 24 hours, so it repeats its motion in a fixed time interval. However, it is not SHM because the conditions for SHM (restoring force directly proportional to displacement) are not met in this context.
2. Motion of an oscillating mercury column in a U-tube: This is nearly simple harmonic motion (SHM). The mercury column oscillates back and forth about its equilibrium position. The restoring force due to gravity and pressure differences is approximately proportional to the displacement, and the motion repeats in a fixed time period.
3. Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point: This is nearly simple harmonic motion (SHM). When released, the ball bearing will oscillate about the lowest point of the bowl. For small displacements, the motion approximates SHM, as the net force acting on the ball is directed towards the equilibrium position and is roughly proportional to the displacement.
4. General vibrations of a polyatomic molecule about its equilibrium position: This is periodic but not simple harmonic motion. Polyatomic molecules have multiple modes of vibration, each with its own frequency. The overall vibration is a complex superposition of many individual simple harmonic motions, making the net motion periodic but not a single SHM.

Summary:

• Simple Harmonic Motion (SHM): (b), (c)
• Periodic but not SHM: (a), (d)

To answer this question definitively, the actual Figure 14.27 is required. However, based on the nature of x-t plots for linear motion, we can describe the criteria for identifying periodic motion and its period.

Criteria for Periodic Motion in an x-t plot:

A plot represents periodic motion if the position 'x' of the particle repeats itself after a fixed time interval. Visually, this means the graph must exhibit a pattern that repeats itself along the time axis.

Identifying the Period (T):

If a plot shows periodic motion, the period (T) is the duration of one complete cycle. This can be found by measuring the horizontal distance between any two consecutive points on the graph that are in the same phase of motion (e.g., two successive peaks, two successive troughs, or two successive points where the particle passes through the equilibrium position moving in the same direction).

General Interpretation of x-t plots:

• A sinusoidal curve (like $x(t)\; =\; A\; \backslash sin(\backslash omega\; t\; +\; \backslash phi)$ or $x(t)\; =\; A\; \backslash cos(\backslash omega\; t\; +\; \backslash phi)$) typically represents simple harmonic motion, which is a form of periodic motion. The period T is related to $\backslash omega$ by $T\; =\; \backslash frac\{2\backslash pi\}\{\backslash omega\}$.
• A graph that is a horizontal line indicates the particle is at rest (not periodic).
• A graph that is a straight line with a non-zero slope indicates constant velocity (not periodic unless it somehow returns, which is unlikely for a single straight line).
• An irregular or non-repeating graph is not periodic.

Conclusion (pending figure):

You would need to examine each of the four plots in Figure 14.27. Any plot that shows a repeating wave-like pattern corresponds to periodic motion. The length of one complete wave cycle along the time axis gives the period.