Calculating New Time Period Of Pendulum After 60% Length Increase
In the realm of physics, the simple pendulum serves as a fundamental concept for understanding oscillatory motion and time period relationships. The time period of a simple pendulum is intrinsically linked to its length and the acceleration due to gravity. This article delves into a specific scenario where we explore how altering the length of a pendulum affects its time period. We will analyze a problem where the initial time period is given as π seconds, and the length is increased by 60%. Our goal is to determine the new time period resulting from this change. Understanding such relationships is crucial for both theoretical physics and practical applications, such as clock mechanisms and other oscillatory systems.
To fully grasp the problem at hand, it's essential to first understand the basic principles governing the motion of a simple pendulum. A simple pendulum, in its idealized form, consists of a point mass suspended from a fixed point by a massless, inextensible string. When displaced from its equilibrium position, the pendulum oscillates back and forth due to the restoring force of gravity. The time period (T) of a simple pendulum, which is the time taken for one complete oscillation, is given by the formula:
T = 2π √(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 m/s²).
This formula reveals that the time period is directly proportional to the square root of the length of the pendulum and inversely proportional to the square root of the acceleration due to gravity. This relationship is pivotal in solving problems involving changes in length or gravitational acceleration.
Our specific problem presents a scenario where the initial time period of a simple pendulum is π seconds. The length of the pendulum is then increased by 60%. The task is to calculate the new time period after this increase in length. This problem requires us to apply the formula for the time period and understand how changes in length affect the oscillation period. We will proceed by first setting up the initial conditions and then calculating the new length and subsequently the new time period.
Setting Up the Initial Conditions
Initially, we are given that the time period T₁ is π seconds. Using the formula for the time period, we can write:
π = 2π √(l₁/g)
Where l₁ is the initial length of the pendulum. From this equation, we can derive a relationship between l₁ and g. This relationship will be crucial in determining how the change in length affects the time period.
Calculating the New Length
The problem states that the length of the pendulum is increased by 60%. This means the new length, l₂, is the initial length plus 60% of the initial length. Mathematically, this can be expressed as:
l₂ = l₁ + 0.60l₁ = 1.60l₁
This new length is the key factor in determining the new time period. By substituting this value into the time period formula, we can see how the time period changes with the increased length.
Determining the New Time Period
Now that we have the new length, l₂, we can calculate the new time period, T₂, using the same formula:
T₂ = 2π √(l₂/g)
Substituting l₂ = 1.60l₁ into this equation, we get:
T₂ = 2π √(1.60l₁/g)
To find the relationship between T₂ and the initial time period T₁, we can use the initial equation π = 2π √(l₁/g). Solving for √(l₁/g) from the initial condition, we get:
√(l₁/g) = π / (2π) = 1/2
Substituting this back into the equation for T₂:
T₂ = 2π √(1.60) * √(l₁/g) = 2π √(1.60) * (1/2) = π √1.60
Simplifying the Result
To simplify the result, we can calculate the square root of 1.60. √1.60 is approximately 1.265. Therefore, the new time period T₂ is:
T₂ ≈ π * 1.265
This means the new time period is approximately 1.265 times the initial time period of π seconds. Thus, the new time period is approximately 1.265π seconds.
To provide a clear understanding, let's walk through the step-by-step calculation:
-
Initial Time Period (T₁):
T₁ = π seconds
π = 2π √(l₁/g)
-
Solve for √(l₁/g):
√(l₁/g) = π / (2π) = 1/2
-
New Length (l₂):
l₂ = l₁ + 0.60l₁ = 1.60l₁ 4. New Time Period (T₂):
T₂ = 2π √(l₂/g)
T₂ = 2π √(1.60l₁/g) 5. Substitute √(l₁/g) = 1/2:
T₂ = 2π √(1.60) * (1/2)
T₂ = π √1.60 6. Calculate √1.60:
√1.60 ≈ 1.265
- Final New Time Period:
T₂ ≈ 1.265π seconds
This detailed calculation ensures that each step is clear and easy to follow, leading to a precise determination of the new time period.
Understanding how changes in length affect the time period of a pendulum has significant implications in various practical applications. Pendulums are used in clocks, metronomes, and other timing devices. The accuracy of these devices depends on the precise control of the pendulum's time period. If the length of the pendulum changes due to temperature variations or other factors, the time period will also change, affecting the accuracy of the device.
For example, in pendulum clocks, the length of the pendulum rod can expand or contract with temperature changes. This affects the clock's accuracy, and clockmakers often use temperature-compensating mechanisms to minimize these effects. Similarly, in metronomes, the adjustable weight changes the effective length of the pendulum, allowing musicians to set the tempo accurately.
Furthermore, the principles governing pendulum motion are also applicable in other oscillatory systems, such as the motion of a swing or the vibrations of a bridge. Understanding the relationship between length and time period helps in designing and analyzing these systems to ensure stability and efficiency.
In summary, we have successfully calculated the new time period of a simple pendulum after its length was increased by 60%. The initial time period was given as π seconds, and by using the formula T = 2π √(l/g), we determined that the new time period is approximately 1.265π seconds. This problem highlights the fundamental relationship between the length of a pendulum and its time period, which is crucial in various applications ranging from clock mechanisms to more complex oscillatory systems.
This analysis not only reinforces the theoretical understanding of simple harmonic motion but also illustrates the practical implications of these concepts in real-world scenarios. By breaking down the problem into manageable steps, we've demonstrated how to approach and solve similar physics problems, emphasizing the importance of a clear and methodical approach in problem-solving.
