Question

In the experiment of simple pendulum we keep \(\theta<5^{\circ}\), so as ensure \(\ldots \ldots \ldots\) (A) mass \(\mathrm{m}\) does not interfere in the time period (B) \(\mathrm{g}\) remains constant (C) the air drag is not too much (D) \(\sin \theta \cong \theta\) where by motion becomes simple harmonic.

Short answer

In the experiment of a simple pendulum, we keep the angle θ less than 5° to ensure that the motion becomes simple harmonic. This is achieved by making a small-angle approximation, which states that \(\sin \theta \cong \theta\), allowing the equation governing the motion of the pendulum to be simplified and treated as simple harmonic motion (Option D).

Step-by-step solution

Mass does not interfere in the time period

We know that the time period of a simple pendulum is given by: T = 2π√(L/g) Here, T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity. Mass (m) is not a factor in this formula. Hence, keeping θ < 5° does not affect the mass interfering in the time period. So, option (A) is not the correct answer. #Step 2: Analyze Option (B)#

Acceleration due to gravity (g) remains constant

The acceleration due to gravity (g) is a constant and is not affected by the angle θ. It might change with altitude, but that is not relevant here. Therefore, option (B) is not the correct answer. #Step 3: Analyze Option (C)#

Air drag is not too much

Air drag does have some effect on the motion of the pendulum. However, the angle θ being less than 5° doesn't significantly affect the air drag, and hence the air drag doesn't really contribute to simple harmonic motion directly. So, option (C) is not the correct answer. #Step 4: Analyze Option (D)#

\(\sin \theta \cong \theta\), and motion becomes simple harmonic

A simple pendulum only exhibits simple harmonic motion when the oscillations are small. In this case, when θ < 5°, we can make a small-angle approximation, which states that: \(\sin \theta \cong \theta\) This approximation allows us to simplify the equation governing the motion of the pendulum, allowing it to be treated as simple harmonic motion. Therefore, option (D) is the correct answer. Thus, the angle must be kept less than 5° in the experiment of a simple pendulum to ensure that the motion becomes simple harmonic (Option D).

Key concepts

Simple Pendulum

A simple pendulum consists of a mass, known as a "bob," attached to the end of a string or rod of length \( L \), which swings back and forth. The movement occurs due to the force of gravity aiming to bring the bob back to its equilibrium position, creating a regular back-and-forth motion. This type of pendulum is used to explore fundamental principles of physics, often in introductory courses or experiments.

• The pendulum's motion is periodic, meaning it repeats itself in a regular time cycle.
• Its length \( L \) and the angle \( \theta \) from which it is released are key factors governing its behavior.
• For simple pendulums, air drag and the mass of the bob are usually considered negligible.

Understanding the simple pendulum lays the groundwork for exploring more complex physical systems.

Small-Angle Approximation

The small-angle approximation is a key concept that simplifies the math involved in pendulum motion. When the angle \( \theta \) is small, specifically less than about \(5^\circ\), the sine of the angle \( (\sin \theta) \) is approximately equal to the angle in radians, \( \theta \). This relationship is expressed as:
\[\sin \theta \approx \theta\]

• This simplification is crucial because it allows us to treat the pendulum's motion as simple harmonic.
• Without this approximation, the pendulum's differential equations become more complex and harder to solve.

Thanks to this approximation, we can predict the pendulum's behavior using simpler models.

Pendulum Time Period

The time period \( T \) of a simple pendulum is the time it takes to complete one full oscillation — moving from one side to the other and back again. The formula for the time period is given by:
\[T = 2\pi\sqrt{\frac{L}{g}}\]
Where:

• \( T \) is the time period.
• \( L \) is the length of the pendulum
• \( g \) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\) on Earth).

Notably, the mass of the bob does not affect the time period.
This surprising result emphasizes the consistent and predictable nature of pendulum motion under uniform gravitational conditions.

Oscillation Angle

The oscillation angle \( \theta \) in a simple pendulum is the initial angle between the string and the vertical. Keeping this angle small is essential for the pendulum to exhibit true simple harmonic motion. Here's why:

• Larger angles would invalidate the small-angle approximation \( \sin \theta \approx \theta \), complicating the motion's analysis.
• Keeping \( \theta < 5^\circ \) ensures minimal error in predictions using simple harmonic models.

Consequently, when conducting experiments, especially those exploring the principles of physics, maintaining a small oscillation angle is critical to achieving accurate and manageable results.