How to Find Acceleration Due to Gravity Using Simple Pendulum

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How to Find Acceleration Due to Gravity Using Simple Pendulum

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In this blog post, we will explore how to find the acceleration due to gravity using a simple pendulum. We will discuss the components and functioning of a simple pendulum, its importance in physics, and a practical experiment to determine the acceleration due to gravity. Additionally, we will delve into the mathematical calculations involved and provide worked-out examples for better understanding.

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A simple pendulum is a weight (or bob) suspended from a fixed point and allowed to swing freely. It consists of a string or rod and is an excellent example of periodic motion. The motion of the bob can be described as oscillation, which involves repetitive motion back and forth around the equilibrium position.

The components of a simple pendulum include:
– The bob: The weight or object at the end of the pendulum.
– The string or rod: The medium that connects the bob to the fixed point.
– The fixed point: The point from which the pendulum is suspended.

Simple pendulums are widely used in physics to study various concepts such as oscillation, angular displacement, periodic motion, and more. They provide a simplified model for understanding more complex systems and have applications in fields like timekeeping, seismology, and physics research.

Practical Experiment to Determine Acceleration Due to Gravity

To determine the acceleration due to gravity using a simple pendulum, you can conduct the following experiment:

Materials Required for the Experiment:

  1. A string or rod of known length (L)
  2. A small weight or bob
  3. A stopwatch or timer
  4. A protractor or angle measuring device

Step-by-Step Procedure to Conduct the Experiment:

  1. Attach the bob to the string or rod and make sure it is securely fastened.
  2. Set the pendulum in motion by pulling the bob to one side and releasing it. Ensure that the motion is smooth and uninterrupted.
  3. Use the stopwatch to measure the time taken for the pendulum to complete a full swing (to and fro). This is known as the period (T) of the pendulum.
  4. Repeat the measurement multiple times to ensure accuracy and calculate the average value of the period.

Safety Measures to Consider During the Experiment:

  • Ensure that the pendulum swings freely without any obstructions.
  • Maintain a safe distance from the pendulum during its motion to avoid any accidents.
  • Handle the weight or bob with care to prevent injury.

Calculating Acceleration Due to Gravity Using Simple Pendulum

To calculate the acceleration due to gravity (g) using a simple pendulum, we can use the following formula:

g = \left(\frac{4\pi^2L}{T^2}\right) )

where:
– g is the acceleration due to gravity
– L is the length of the pendulum
– T is the period of the pendulum

Let’s understand this formula better with the help of a worked-out example:

Worked Out Example:

Suppose we have a simple pendulum with a length of 1 meter (L = 1m) and a period of 2 seconds (T = 2s). To find the acceleration due to gravity, we can use the formula:

g = \left(\frac{4\pi^2L}{T^2}\right) )

Substituting the given values:

g = \left(\frac{4\pi^2 \times 1}{2^2}\right) )

Simplifying the equation:

g = \frac{4\pi^2}{4} )

g = \pi^2 )

Hence, the acceleration due to gravity in this scenario is approximately \pi^2 ) m/s^2

Common Mistakes to Avoid While Calculating

While calculating the acceleration due to gravity using a simple pendulum, it is important to avoid the following common mistakes:
– Forgetting to square the period (T^2) in the formula.
– Using the wrong length (L) of the pendulum.
– Using an incorrect value of pi (π). Remember to use the accurate value according to your calculation requirements.

By following the correct formula and taking accurate measurements, you can ensure reliable results and minimize errors in your calculations.

Numerical Problems on how to find acceleration due to gravity using simple pendulum

Problem 1:

A simple pendulum of length 1.5 m is oscillating with a period of 2 seconds. Determine the acceleration due to gravity.

Solution:

Given:
Length of the pendulum (L) = 1.5 \, \text{m}
Time period (T) = 2 \, \text{s}

The formula relating the period of a simple pendulum to the length and acceleration due to gravity is:

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

To find the acceleration due to gravity (g), we rearrange the formula as follows:

g = \frac{4\pi^2 L}{T^2}

Substituting the given values into the formula, we get:

g = \frac{4 \times \pi^2 \times 1.5}{(2)^2}

Simplifying further:

g = \frac{9 \pi^2}{2} \approx 44.17 \, \text{m/s}^2

Therefore, the acceleration due to gravity is approximately 44.17 m/s^2.

Problem 2:

A simple pendulum of length 2 m is oscillating with a period of 3 seconds. Find the value of acceleration due to gravity.

Solution:

Given:
Length of the pendulum (L) = 2 \, \text{m}
Time period (T) = 3 \, \text{s}

Using the formula for the period of a simple pendulum, we have:

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

Solving for g, we get:

g = \frac{4\pi^2 L}{T^2}

Substituting the given values:

g = \frac{4 \times \pi^2 \times 2}{3^2}

Simplifying further:

g = \frac{8 \pi^2}{9} \approx 8.85 \, \text{m/s}^2

Therefore, the acceleration due to gravity is approximately 8.85 m/s^2.

Problem 3:

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A simple pendulum of length 0.8 m completes 20 oscillations in 30 seconds. Calculate the acceleration due to gravity.

Solution:

Given:
Length of the pendulum (L) = 0.8 \, \text{m}
Number of oscillations (n) = 20
Time taken (t) = 30 \, \text{s}

The formula connecting the number of oscillations, time taken, and period of a simple pendulum is:

T = \frac{t}{n}

Using the formula for the period of a simple pendulum, we have:

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

Combining the above two equations, we can solve for g:

g = \frac{4\pi^2 L}{T^2}

Substituting the values:

g = \frac{4\pi^2 \times 0.8}{\left(\frac{30}{20}\right)^2}

Simplifying further:

g = \frac{256 \pi^2}{225} \approx 35.73 \, \text{m/s}^2

Therefore, the acceleration due to gravity is approximately 35.73 m/s^2.

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