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Simple Pendulum: Definition, Formula, and Calculations


When it comes to understanding the principles of physics, the simple pendulum serves as an excellent starting point. This fundamental concept helps students and enthusiasts grasp the basics of oscillatory motion and the principles that govern it. In this article, we will explore what a simple pendulum is, its formula, and delve into the calculations involved. We will break down the concepts into easily digestible explanations, empowering you to master this topic with ease.

What is a Simple Pendulum?

A simple pendulum is a basic physics apparatus consisting of a mass (known as the pendulum bob) suspended from a fixed point by a string or rod. It is a classic example of periodic motion, where the pendulum swings back and forth in a regular and repeating pattern.

The Simple Pendulum Formula

To understand the behavior of a simple pendulum, we use the formula that relates the period of oscillation to its length. The formula for the period (T) of a simple pendulum is given as:

T = 2π √(L / g)


  • T is the period of the pendulum (time it takes for one complete swing).
  • L is the length of the pendulum from the pivot point to the center of mass of the bob.
  • g is the acceleration due to gravity.

Calculating the Period of a Simple Pendulum

To calculate the period of a simple pendulum, follow these steps:

  1. Measure the length of the pendulum (L) from the pivot point to the center of mass of the pendulum bob using a ruler or measuring tape.
  2. Determine the acceleration due to gravity (g) at your location. For most practical purposes on the Earth’s surface, the average value is approximately 9.81 m/s².
  3. Plug the values of L and g into the simple pendulum formula:
    T = 2π √(L / g)

Let’s take an example to illustrate the calculation. Suppose the length of the pendulum (L) is 0.5 meters, and the acceleration due to gravity (g) is 9.81 m/s²:

T = 2π √(0.5 / 9.81)


T ≈ 2π √0.051 (rounded to three decimal places)

We will now have

T ≈ 2π × 0.226

And finally

T ≈ 1.419 seconds (rounded to three decimal places)

Hence, the period of this simple pendulum is approximately 1.419 seconds.

Frequency of a Simple Pendulum

The frequency (f) of a simple pendulum represents the number of oscillations it completes in one second. It is the reciprocal of the period where the formula is:

f = 1 / T

where T is the period of the pendulum.

Using the previous example, where the period (T) was calculated to be approximately 1.419 seconds:

f = 1 / 1.419

f ≈ 0.705 Hz (rounded to three decimal places)

Therefore, the frequency of the simple pendulum in this example is approximately 0.705 Hz.

Amplitude and Displacement

The amplitude of a simple pendulum refers to the maximum angle it reaches during its swing. It is directly related to the initial displacement of the pendulum from its equilibrium position.

For small angles (up to around 15 degrees), the period remains constant and can be calculated using the formula mentioned earlier. However, for larger angles, the pendulum’s motion becomes more complex, and the simple pendulum formula may not accurately predict its period.

Factors Affecting the Period of a Simple Pendulum

Several factors can influence the period of a simple pendulum:

  1. Length (L) of the Pendulum: The longer the length, the longer the period. As the length increases, the pendulum takes more time to complete one full swing.
  2. Acceleration due to Gravity (g): Different locations on Earth have slightly different values of g, affecting the period of the pendulum.
  3. Amplitude: For small angles, the period remains constant, but as the amplitude increases, the period may vary.
  4. Air Resistance: In real-world scenarios, air resistance can dampen the motion of the pendulum, affecting its period.

Advantages of Using Simple Pendulums

Simple pendulums have several advantages in various applications:

  • Timekeeping: Historically, simple pendulums were used in clocks to maintain accurate timekeeping.
  • Education: They serve as excellent educational tools to teach concepts of periodic motion and mechanics.
  • Gravity Measurement: Measuring the period of a pendulum allows for the calculation of local gravity, aiding geophysical studies.


Q: Does the mass of the pendulum bob affect its period?
A: Surprisingly, the mass of the pendulum bob does not affect its period. As long as the length remains the same, the mass has no impact on the time it takes for the pendulum to complete one swing.

Q: Can a simple pendulum be used to measure gravitational acceleration?
A: Yes, a simple pendulum can be utilized to calculate the acceleration due to gravity. By measuring the period and length of the pendulum, one can derive the local value of gravitational acceleration.

Q: What happens if the length of the pendulum is halved?
A: Halving the length of the pendulum will result in a shorter period, meaning the pendulum will complete its swings in less time.

Q: Can a simple pendulum swing indefinitely without any external force?
A: In an ideal scenario without air resistance or friction, a simple pendulum would continue to swing indefinitely, preserving its mechanical energy.

Q: Is a simple pendulum an example of harmonic motion?
A: Yes, a simple pendulum exhibits harmonic motion since its motion is governed by a restoring force (gravity) that is proportional to its displacement from the equilibrium position.

Q: Are there any real-world applications of the simple pendulum formula?
A: Absolutely! The simple pendulum formula finds applications in various fields, such as physics, engineering, and even seismology, where it aids in understanding harmonic motion and vibrations.


Understanding the simple pendulum, its formula, and calculations is a gateway to comprehending the principles of periodic motion and the underlying physics concepts. By mastering this fundamental topic, learners can build a solid foundation for more complex mechanical and oscillatory systems. Whether you are a student, educator, or simply curious about the world around you, delving into the intricacies of the simple pendulum is a fascinating journey.

Next time you encounter a grandfather clock or a swinging amusement park ride, you’ll have a deeper appreciation for the mechanics behind them. Embrace the wonder of oscill

ation and let the simple pendulum guide you through the captivating world of periodic motion.