Understanding the Simple Pendulum and Time Period

A simple pendulum is a fundamental concept in physics used to study oscillatory motion. It consists of a point mass, often called a bob, suspended by a light, inextensible string from a fixed support. When the bob is pulled to one side and released, it swings back and forth due to gravity, exhibiting periodic motion.

The time period of a simple pendulum is defined as the time taken for one complete oscillation (a swing from one extreme position to the other and back to the starting extreme position).

Calculating the Time Period of a 1m Pendulum

The time period ($T$) of a simple pendulum for small angular displacements is determined by its length ($L$) and the acceleration due to gravity ($g$) at the location. The formula that describes this relationship is:

$T = 2\pi \sqrt{\frac{L}{g}}$

Where:

• $T$ is the time period in seconds (s).
• $\pi$ is the mathematical constant pi (approximately 3.14159).
• $L$ is the length of the pendulum in meters (m).
• $g$ is the acceleration due to gravity in meters per second squared (m/s²). On the surface of the Earth, the average value of $g$ is approximately $9.8 \, m/s^2$.

Step-by-Step Calculation for L = 1m

We are given that the length of the pendulum is $L = 1 \, m$. We will use the standard value for the acceleration due to gravity, $g \approx 9.8 \, m/s^2$.

Substitute these values into the formula for the time period:

$T = 2\pi \sqrt{\frac{1 \, m}{9.8 \, m/s^2}}$

First, calculate the value inside the square root:

$\frac{1}{9.8} \approx 0.10204 \, s^2$

Now, take the square root:

$\sqrt{0.10204 \, s^2} \approx 0.3194 \, s$

Finally, multiply by $2\pi$ (using $\pi \approx 3.14159$):

$T \approx 2 \times 3.14159 \times 0.3194 \, s$

$T \approx 6.28318 \times 0.3194 \, s$

$T \approx 2.006 \, s$

Comparing Calculated Time Period with Options

The calculated time period for a 1 m long pendulum is approximately 2.006 seconds. Let's look at the provided options:

• Option 1: 6 s
• Option 2: 4 s
• Option 3: 2 s
• Option 4: 1 s

The calculated value $2.006 \, s$ is extremely close to 2 seconds. Therefore, the time period of a 1 m long pendulum approximates to 2 seconds.

Revision Table: Simple Pendulum Time Period Factors

Additional Information on Pendulum Physics

The formula used for the time period of a simple pendulum assumes that the oscillations are small (amplitude is small). When the amplitude is large, the motion is not perfectly simple harmonic, and the actual time period is slightly longer than that given by the formula.

The value of $g$ is not constant across the Earth's surface. It varies slightly with latitude, altitude, and local geological structure. This means that a pendulum clock would run at a slightly different rate if moved from one location to another with a different value of $g$.

For practical pendulums (physical pendulums) which have distributed mass rather than a point mass bob, the time period calculation involves the moment of inertia about the pivot point and the distance to the center of mass.