Problem 23 - Basic Engineering Mathematics

Question

The time of swing \(t\) seconds, of a simple pendulum is given by \(t=2 \pi \sqrt{\frac{l}{g}}\). Determine the time, correct to 3 decimal places, given that \(l=12.0\) and \(g=9.81\).

Step-by-Step Solution

Answer

The time is 6.932 seconds.

Step 1: Understanding the Formula

The formula for the time of swing of a simple pendulum is given by \( t = 2 \pi \sqrt{\frac{l}{g}} \). Here, \( l \) is the length of the pendulum, and \( g \) is the acceleration due to gravity.

Step 2: Plugging in the Values

Substitute the given values into the formula, where \( l = 12.0 \) and \( g = 9.81 \). Thus, equation becomes, \[ t = 2 \pi \sqrt{\frac{12.0}{9.81}} \].

Step 3: Calculating the Square Root

First, calculate the fraction inside the square root: \( \frac{12.0}{9.81} \). This results in approximately \( 1.22 \). Then, find the square root of \( 1.22 \), which is approximately \( 1.104 \).

Step 4: Calculating the Time

Multiply the results by \( 2\pi \): \( t = 2 \times 3.1416 \times 1.104 \). This gives \( t \approx 6.932 \).

Step 5: Rounding to the Required Decimal Places

Finally, round off \( 6.932 \) to three decimal places. The time of swing \( t \) is \( 6.932 \) seconds.

Key Concepts

Simple Pendulum Formula

The simple pendulum formula is a fundamental physics equation that describes the motion of a simple pendulum. A simple pendulum consists of a weight, or bob, attached to the end of a string or rod of fixed length, which swings back and forth under the influence of gravity. The formula is given by:

\[ t = 2 \pi \sqrt{\frac{l}{g}} \]

• \(t\) represents the time it takes for one complete oscillation, known as the period.
• \(l\) is the length of the pendulum.
• \(g\) denotes the acceleration due to gravity, typically 9.81 m/s² on Earth's surface.

This formula helps to predict how long a pendulum will take to swing back and forth, based solely on its length and the gravitational pull. It does not depend on the mass of the bob or the amplitude of the swing, provided the angle is small.

Length of Pendulum

The length of the pendulum, denoted by \(l\) in the formula, is a crucial factor in determining the time period of the pendulum's swing. The length is the measurement from the pivot point to the center of mass of the pendulum's bob.

A longer pendulum means a longer period, and therefore, a slower swing. This is because:

• A longer pendulum travels a greater distance during each oscillation.
• The speed at which the pendulum moves is lower for longer lengths, given the same gravitational pull.

For instance, in our exercise, the pendulum length is 12.0 meters. By substituting this value into our formula, we can calculate how the length influences the pendulum period.

Acceleration Due to Gravity

The acceleration due to gravity, \(g\), is the rate at which an object accelerates when it is in free fall towards the Earth without any air resistance. On our planet, this value is approximately 9.81 m/s².

It plays a crucial role in determining the swing period of a pendulum. A stronger gravitational pull (higher \(g\) value) results in a shorter swing period, while a weaker gravitational pull results in a longer period.

• The gravitational force affects how quickly the pendulum is pulled back toward its resting position.
• It is a constant value under similar conditions at the same location, simplifying calculations.

In our exercise, we used \(g = 9.81\) which is standard for calculations taken on Earth.

Calculation of Pendulum Period

To calculate the pendulum's period, we must follow a simple substitution and calculation process using the variables discussed.

Steps for calculation in the exercise involve:

• Substitute the length \(l = 12.0\) meters and \(g = 9.81\) in the swing period formula: \[ t = 2 \pi \sqrt{\frac{12.0}{9.81}} \]
• First, calculate the fraction \( \frac{12.0}{9.81} \), resulting in approximately 1.22.
• Find the square root of 1.22, which is approximately 1.104.
• Multiply 1.104 by \(2\pi\) (approximately 6.283), giving a period \(t \approx 6.932\) seconds.

Finally, round to three decimal places to get the period of the pendulum, which is \(6.932\) seconds. This method shows the direct relationship between the pendulum's length, gravity, and its period.