Question

The distance travelled by a particle performing S.H.M. during time interval equal to its periodic time is \(\ldots \ldots\) (A) A (B) \(2 \mathrm{~A}\) (C) \(4 \mathrm{~A}\) (D) Zero.

Short answer

The distance traveled by a particle performing simple harmonic motion (SHM) during a time interval equal to its periodic time is \(2 \times A\). The correct answer is (B) \(2 \mathrm{~A}\).

Step-by-step solution

Recall the properties of SHM

Simple harmonic motion (SHM) is a type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. Some important properties of SHM are: 1. The particle oscillates between two extreme points symmetrically placed around the mean position. 2. The maximum displacement from the mean position is called the amplitude (A). 3. The time taken by the particle to complete one full oscillation is called the periodic time (T).

Understand the meaning of periodic time

Periodic time (T) is the time taken by the particle to complete one full oscillation or cycle. It is also the time interval after which the motion of the particle repeats itself in a periodic manner. The particle will be back to its initial position after completing one cycle.

Calculate the distance traveled during the periodic time

During one cycle of periodic time, the particle moves from one extreme position to the other and back to the initial extreme position. Since the amplitude represents the maximum displacement from the mean position, during one cycle, the particle will cover a distance equal to 2 times the amplitude (once while going from one extreme to the other and once while returning to its initial position). Therefore, the distance traveled by a particle performing SHM during a time interval equal to its periodic time is \(2 \times A\). The correct answer is (B) \(2 \mathrm{~A}\).

Key concepts

Amplitude

The amplitude in Simple Harmonic Motion (SHM) refers to the maximum extent of displacement from the equilibrium position. It is a crucial factor that determines how far the oscillating object moves from its central point. Understanding amplitude helps to grasp the full scope of oscillations.

Here are a few key points about amplitude:

• Maximum Displacement: Amplitude is the maximum distance the object moves from its rest position.
• Symbol: It is usually denoted by \( A \) and is measured in units of length, such as meters or centimeters.
• Energy Component: Amplitude is directly related to the energy in the oscillating system; a greater amplitude indicates more energy involved.

Amplitude provides a clear measure of how vigorous the motion is, offering insights into the system's energy and stability. It's a foundational concept when studying various oscillatory movements, not just limited to SHM.

Periodic Time

Periodic time, often symbolized by \( T \), is the duration required for an oscillating particle to complete one full cycle of motion. Understanding periodic time is essential for analyzing how frequently an oscillation occurs.

Let's take a closer look at periodic time:

• Complete Cycle: It is the time it takes for the particle to return to its original position and state after an oscillation.
• Measurements: Periodic time is measured in seconds, and it reflects the frequency of oscillations. High frequency corresponds to a short periodic time and vice versa.
• Repetition of Motion: Periodic time underscores the repetitive nature of the motion, making it predictable and manageable to study.

Grasping the concept of periodic time is crucial in fields such as physics and engineering, where understanding the timing of oscillations is vital for applications ranging from clocks to musical instruments.

Oscillatory Motion

Oscillatory motion characterizes a type of motion where an object moves back and forth in a regular pattern about a central position. Simple Harmonic Motion (SHM) is a classic example of oscillatory motion, with distinct features that define its behavior.

Here are the significant aspects of oscillatory motion:

• Restoring Force: In SHM, this force is proportional and opposite to the displacement, guiding the particle back towards the equilibrium position.
• Symmetry: The motion is always symmetric around a central, mean position which it frequently returns to.
• Predictability: Oscillatory motion is highly predictable due to its repetitive nature, allowing for the calculation and prediction of future states.

Understanding oscillatory motion is pivotal in various scientific and engineering disciplines, revealing how systems stabilize and maintain equilibrium over time. This understanding aids in the design and analysis of everything from bridges to electronic circuits.