Question

Complete the following sentence. Time period of oscillation of a simple pendulum is dependent on......... (A) Length of thread (B) initial phase (C) amplitude (D) mass of bob

Short answer

Time period of oscillation of a simple pendulum is dependent on the Length of thread (A).

Step-by-step solution

Recall the formula for the time period of a simple pendulum

The time period of oscillation (T) of a simple pendulum is given by the formula: \(T = 2\pi\sqrt{\frac{l}{g}}\) Where: - T is the time period - l is the length of the pendulum (length of the thread) - g is the acceleration due to gravity

Analyze the dependencies

According to the formula, the time period of oscillation (T) depends on: 1. Length of the pendulum (l) 2. Acceleration due to gravity (g) Notice that the initial phase, amplitude, and mass of the bob are not present in the formula for the time period. Thus, these factors do not affect the time period of a simple pendulum.

Choose the correct option

Based on our analysis, only option (A) Length of thread is a factor that affects the time period of oscillation of a simple pendulum. Therefore, the correct answer is: Time period of oscillation of a simple pendulum is dependent on the Length of thread (A).

Key concepts

Time Period of Oscillation

The time period of oscillation refers to the time it takes for one complete cycle of motion in a simple pendulum. To understand this better, imagine a merry-go-round, where the time period is the duration for one full rotation. For a simple pendulum, this period can be calculated using the formula:\[T = 2\pi\sqrt{\frac{l}{g}}\]- **T** is the time period.- **l** is the length of the pendulum.- **g** is the acceleration due to gravity.The relationship between these elements means that the time it takes for the pendulum to swing back and forth is not random. Instead, it is systematically determined by factors like length and gravity. Notably, aspects such as initial phase, amplitude, and mass do not come into play here. Thus, if you were to change the amplitude or mass, without changing the length or gravity (which remains constant on Earth), the time period would remain unaffected. This is only true for small amplitude swings, typical for small angles in a simple pendulum.

Length of Pendulum

The length of the pendulum is a critical factor influencing the time period of oscillation. Specifically, this is the length from the point of suspension to the center of mass of the pendulum's bob. If you were to increase the length of the pendulum, the time period would increase too. This is because the pendulum takes a longer path and thus needs more time to complete one oscillation. Imagine drawing a large circle versus a small circle. A line traveling the perimeter of the larger circle has a longer distance to cover compared to the smaller one. Similarly, a longer pendulum covers more distance. Here's what you need to remember:

• A longer pendulum means a longer time period.
• A shorter pendulum means a shorter time period.

This inversely affects the frequency of oscillation, where frequency is the number of cycles per unit of time. So, as the length increases, the frequency decreases since each oscillation takes more time.

Acceleration Due to Gravity

Acceleration due to gravity is another essential component of the pendulum's time period formula. Here on Earth, we use the average value for gravity, approximately 9.81 m/s². In the formula, gravity affects how quickly the pendulum accelerates back towards its equilibrium position during its oscillation cycle. Consider what happens if you were to take the pendulum to a different planet where gravity is stronger or weaker:

• Stronger gravity (larger g value) would result in a shorter time period. This is because the pendulum is pulled back more forcefully, making it swing faster.
• Weaker gravity (smaller g value) would result in a longer time period, since the pendulum would swing more slowly.

Therefore, the value of gravitational acceleration is inversely proportional to the time it takes for one complete cycle. This showcases how the environment affects pendulum dynamics, providing insights into not just physics, but also planetary science and engineering.