Animated Solution for Physics - Oscillations: If the time period of a 2 m long simple pendulum is 2 s, the acceleration due to gravity at the place, where pendulum is executing SHM is

The simple pendulum is one of the most elegant and fundamental systems in physics. It beautifully connects the concepts of kinematics, forces, and periodic motion. In this problem, we are going to use the motion of a pendulum to deduce the local acceleration due to gravity. Let's dive into the mechanics of this setup!

Imagine a simple pendulum swinging back and forth in a steady, rhythmic motion. We are given two crucial pieces of information about this system:

1. The length of the pendulum string, $l=2\text{ m}$. 2. The time it takes to complete one full oscillation, known as the time period, $T=2\text{ s}$.

Our objective is to find the effective acceleration due to gravity, $g_{\text{eff}}$, at the specific location where this pendulum is executing Simple Harmonic Motion (SHM).

To bridge the gap between the length, the time period, and gravity, we rely on the standard formula for the time period of a simple pendulum:

This equation tells us that the time period is directly proportional to the square root of the length and inversely proportional to the square root of the effective gravity.

Since our goal is to find $g_{\text{eff}}$, we need to isolate it. The first step is to eliminate the square root by squaring both sides of the equation:

Now, we can easily rearrange this equation to make $g_{\text{eff}}$ the subject. By multiplying both sides by $g_{\text{eff}}$ and dividing by $T^2$, we get:

With our formula perfectly arranged, it's time to plug in the numbers. We substitute $l=2$ and $T=2$ into our equation:

Let's simplify the math. In the denominator, $2^2$ becomes $4$.

The $8$ divided by $4$ simplifies neatly to $2$, leaving us with our final result:

You might wonder why we use the term $g_{\text{eff}}$ instead of just $g$. This is a crucial distinction in physics! If this pendulum were placed inside an elevator accelerating upwards, the bob would feel heavier, and the effective gravity would increase ($g_{\text{eff}}=g+a$). Conversely, if the elevator accelerated downwards, the effective gravity would decrease ($g_{\text{eff}}=g-a$). Always pay attention to the frame of reference when dealing with pendulum problems!