Calculating Circular Disc Revolutions

This problem involves calculating the number of revolutions a circular disc makes while rolling a certain distance, given its area. To solve this, we first need to find the radius of the disc from its area, then calculate its circumference, and finally divide the total distance rolled by the circumference to find the number of revolutions.

Understanding the Concepts

• Area of a Circle: The area (A) of a circle is given by the formula \(A = \pi r^2\), where \(r\) is the radius.
• Circumference of a Circle: The circumference (C) of a circle (or a disc's edge) is given by the formula \(C = 2\pi r\). This represents the distance covered in one full revolution when the disc rolls.
• Number of Revolutions: When a disc rolls without slipping, the distance covered in one revolution is equal to its circumference. Therefore, the total number of revolutions is the total distance rolled divided by the circumference.

Step-by-Step Calculation

Step 1: Find the Radius of the Circular Disc

We are given the area of the circular disc as \(0.64 \pi \, \text{m}^2\). We use the area formula to find the radius (\(r\)).

Area \(A = \pi r^2\)

Given \(A = 0.64 \pi \, \text{m}^2\)

So, \(0.64 \pi = \pi r^2\)

Divide both sides by \(\pi\):

\(0.64 = r^2\)

Take the square root of both sides:

\(r = \sqrt{0.64} = 0.8 \, \text{m}\)

The radius of the circular disc is 0.8 meters.

Step 2: Calculate the Circumference of the Circular Disc

Now that we have the radius, we can calculate the circumference (\(C\)) using the formula \(C = 2\pi r\). We are asked to take \(\pi = 22/7\).

\(C = 2 \times \frac{22}{7} \times 0.8\)

\(C = \frac{44}{7} \times 0.8\)

\(C = \frac{44 \times 0.8}{7}\)

\(C = \frac{35.2}{7} \, \text{m}\)

The circumference of the circular disc is \(35.2/7\) meters.

Step 3: Convert the Total Distance Rolled to Meters

The total distance rolled is given as 1.408 km. We need to convert this distance into meters, as the circumference is in meters (1 km = 1000 m).

Distance = \(1.408 \, \text{km}\)

Distance = \(1.408 \times 1000 \, \text{m}\)

Distance = \(1408 \, \text{m}\)

The total distance rolled is 1408 meters.

Step 4: Calculate the Number of Revolutions

The number of revolutions is the total distance rolled divided by the circumference per revolution.

Number of revolutions = \(\frac{\text{Total Distance}}{\text{Circumference}}\)

Number of revolutions = \(\frac{1408 \, \text{m}}{35.2/7 \, \text{m}}\)

Number of revolutions = \(1408 \times \frac{7}{35.2}\)

Number of revolutions = \(\frac{1408 \times 7}{35.2}\)

Let's perform the multiplication in the numerator:

\(1408 \times 7 = 9856\)

Number of revolutions = \(\frac{9856}{35.2}\)

To divide by 35.2, we can multiply the numerator and denominator by 10 to remove the decimal:

Number of revolutions = \(\frac{98560}{352}\)

Performing the division:

Number of revolutions = \(280\)

The number of revolutions the circular disc makes is 280.

Summary of the Solution

We started with the area of the circular disc to find its radius. Using the radius, we calculated the circumference, which is the distance covered in one revolution. Finally, we divided the total distance rolled by the circumference to find the total number of revolutions.

Revision Table: Circular Motion Calculations

Additional Information: Rolling Motion

When a circular object like a disc or wheel rolls without slipping on a surface, a specific relationship exists between its linear motion and its rotational motion. The distance covered by the center of the object is equal to the length of the arc segment of the circumference that has touched the surface. For one full revolution, this distance is exactly equal to the circumference of the object.

This principle is fundamental in understanding how wheels work and is applied in various calculations involving vehicles, gears, and other rotating machinery.