Calculating Pendulum Length and Period: A Practical Guide Using 1.5 Hz

Understanding the behavior of a simple pendulum is crucial in various scientific and engineering applications. In this guide, we will walk through the steps to calculate the periodic time and the length of a pendulum given its frequency. Specifically, we will use a frequency of 1.5 Hz to perform these calculations.

1. Determining the Periodic Time

The periodic time (T) of a simple pendulum can be directly calculated from its frequency (f) using the formula:

[ T frac{1}{f} ]

Given the frequency (f 1.5 ,text{Hz}), the periodic time can be calculated as:

[ T frac{1}{1.5} approx 0.67 ,text{s} ]

This means that a simple pendulum with a frequency of 1.5 Hz will have a period of approximately 0.67 seconds.

2. Determining the Length of the Pendulum

To determine the length of the pendulum, we use the formula for the period of a simple pendulum:

[ T 2pi sqrt{frac{L}{g}} ]

Where:

(T) is the period in seconds (L) is the length of the pendulum in meters (g) is the acceleration due to gravity, approximately (9.8 ,text{m/s}^2)

Rearranging the formula to solve for (L), we get:

[ L frac{g T^2}{4pi^2} ]

Substituting the known values:

[ g 9.8 ,text{m/s}^2 ]

[ T approx 0.67 ,text{s} ]

First, we need to calculate:

[ 0.67^2 approx 0.4489 ]

Now, substituting the values into the equation for (L):

[ L approx frac{9.8 times 0.4489}{4pi^2} ]

Calculating (4pi^2) gives us:

[ 4pi^2 approx 39.478 ]

Thus:

[ L approx frac{4.394}{39.478} approx 0.111 ,text{m} ]

This means the length of the pendulum is approximately:

[ 0.111,text{m} text{ or } 11.1,text{cm} ]

3. Relevant Formulas and Relationships

The relationship between the period of a pendulum (T) and its frequency (f) is:

[ T frac{1}{f} ]

For a simple pendulum, the period (T) can also be determined using the formula:

[ T 2pi sqrt{frac{L}{g}} ]

Rearranging to find the length (L) gives us:

[ L frac{g T^2}{4pi^2} ]

4. Conclusion

In summary, using a frequency of 1.5 Hz, we calculated that the periodic time of the pendulum is approximately 0.67 seconds, and the length of the pendulum is approximately 0.111 meters (or 11.1 cm).

This guide has provided a step-by-step explanation of the calculations involved, utilizing the fundamental physics principles underlying the behavior of simple pendulums.