Engineering Physics by HK Malik

Explanation: In simple harmonic motion, certain quantities help describe the exact condition of the vibrating particle at any instant. One quantity represents the initial setting of the motion at time zero, while another keeps changing as time passes and indicates the current stage of Oscillation. These ideas are important in understanding waves, vibrations, and Periodic motion.

The phase of a particle is linked with angular frequency and time, so it changes continuously as the particle oscillates. It determines whether the particle is near the mean position, extreme position, or somewhere in between. On the other hand, the epoch is connected with the initial phase constant and is fixed for a given motion unless external conditions change.

A useful way to think about this is to compare SHM with the hand of a clock. The starting orientation of the hand is like epoch, while the continuously rotating position of the hand represents phase. One stays as the reference setting, while the other changes every moment.

Thus, understanding the difference between these two quantities requires identifying which parameter remains constant and which evolves with time during Oscillation.

Explanation: When the liquid in a U-shaped tube is disturbed slightly and released, the liquid column performs simple harmonic motion due to the restoring force produced by gravity. The displaced liquid tries to return to equilibrium, and because of inertia, it continues oscillating about the mean position.

The time period of Oscillation of a liquid column in a U-tube depends mainly on the effective length of the liquid column and acceleration due to gravity. The relation resembles the standard SHM formula where the restoring force is proportional to displacement. Here, the total liquid length participating in motion is important rather than the displacement itself.

The Oscillation becomes slower for larger liquid lengths because the moving Mass increases. Gravity acts as the restoring agent that continuously pulls the liquid back toward equilibrium. By substituting the given liquid height into the SHM relation for a U-tube, the required Oscillation time can be estimated accurately.

This motion is similar to a swing moving back and forth after a small push. The system repeatedly exchanges kinetic and potential energy while oscillating around the equilibrium position.

Explanation: In simple harmonic motion, the displacement of a particle changes sinusoidally with time. The particle moves fastest at the mean position and gradually slows as it approaches the extreme positions. To determine the time required to reach a certain fraction of the amplitude, the displacement equation of SHM must be used carefully.

The displacement in SHM is commonly represented using sine or cosine functions involving angular frequency and time. Since the particle starts from the mean position, the sine form becomes especially useful. The given displacement corresponds to a fixed fraction of the amplitude, allowing the trigonometric ratio to be identified directly.

Angular frequency is connected with the time period through a standard relation. After finding the corresponding angle for the required displacement, the time can be calculated from the phase relation. This method shows how SHM links circular motion ideas with oscillatory motion through trigonometric behavior.

A rotating wheel provides a useful analogy. The shadow of a point moving uniformly in a circle behaves like SHM, and reaching half the maximum height corresponds to a specific angular rotation.

Explanation: A phase difference in oscillatory motion describes how much one oscillating particle leads or lags another during vibration. When two pendulums differ in phase by 90°, their positions and velocities are shifted by one-quarter of a complete Oscillation cycle.

In simple harmonic motion, maximum speed occurs when the particle passes through the mean position because the restoring force becomes zero there. At the extreme position, however, the particle momentarily stops before reversing direction. A phase difference of 90° means that when one pendulum is at one important stage of motion, the other occupies another corresponding stage separated by a quarter cycle.

To analyze the situation, imagine the motion represented on a circle. If pendulum A is crossing the equilibrium position with greatest speed, pendulum B will be located at a position shifted by one-quarter rotation. At that instant, its motion characteristics become very different from those of pendulum A.

This relationship is similar to two dancers moving in rhythm but starting at different times. When one reaches the center of motion rapidly, the other reaches a turning point in the cycle.