Problem 21 - Applied Physics

Question

A wheel \(30.0 \mathrm{~cm}\) in diameter moving along level ground made 145 complete rotations. How many metres did the wheel travel?

Step-by-Step Solution

Answer

The wheel traveled approximately 136.30 meters.

Step 1: Find the Radius of the Wheel

The diameter of the wheel is given as 30.0 cm. The radius is half of the diameter, which is calculated as follows:\[ r = \frac{d}{2} = \frac{30.0 \text{ cm}}{2} = 15.0 \text{ cm} \]

Step 2: Convert Radius to Metres

To calculate the travel distance in meters, it's essential to convert the radius from centimeters to meters. Since 1 meter is 100 centimeters:\[ r = \frac{15.0 \text{ cm}}{100} = 0.15 \text{ m} \]

Step 3: Calculate the Circumference of the Wheel

The circumference of a circle is given by the formula \( C = 2\pi r \). Using the radius in meters:\[ C = 2 \times \pi \times 0.15 \text{ m} \approx 0.94 \text{ m} \]

Step 4: Calculate the Total Distance Traveled

The total distance traveled by the wheel is the circumference multiplied by the number of rotations. Therefore:\[ \text{Distance} = 145 \times 0.94 \text{ m} \approx 136.30 \text{ m} \]

Key Concepts

Radius Conversion

In geometry, the radius is a key property of a circle. Simply put, the radius is the distance from the center of the wheel to its edge. When given the diameter, which is the total distance from one side of the circle to the other through the center, the radius is half of that measure. In our exercise, the diameter is given as 30.0 cm, so the radius is calculated as follows:

• The formula to find the radius from the diameter is: \( r = \frac{d}{2} \)
• For the given diameter, substitute: \( r = \frac{30.0 \text{ cm}}{2} = 15.0 \text{ cm} \)

To aid in further calculations, it’s important to convert this into the correct units needed. After all, working in meters is necessary for understanding the wheel’s travel distance clearly.

Circumference Formula

The circumference of a circle is the total distance around it. This is significant when calculating how far a wheel has traveled after multiple rotations. The formula to calculate circumference when you know the radius is:

• \( C = 2\pi r \)

In this formula, \( \pi \) is a constant approximately equal to 3.14159. The exercise requires calculating the circumference using the radius in meters. So once you have:

• Converted 15.0 cm radius to 0.15 m

You can calculate the wheel's circumference in meters:

• \( C = 2 \times \pi \times 0.15 \text{ m} \approx 0.94 \text{ m} \)

By knowing how far a single rotation takes the wheel (the circumference), you can determine how far multiple rotations will take it.

Unit Conversion

Unit conversion is essential when working with measurements of different units. In this scenario, we needed to convert the wheel's radius from centimeters to meters for consistency and ease in distance calculations. Here is how simple unit conversion works:

• 100 centimeters = 1 meter
• Therefore, to convert from cm to m, divide by 100.

From the exercise:

• The radius of 15.0 cm was converted to meters: \( \frac{15.0 \text{ cm}}{100} = 0.15 \text{ m} \)

It is critical, especially in physics and engineering, to use consistent units to prevent errors and ensure accurate calculations. Hence, converting all related measurements to the same unit of measure before proceeding with problem-solving is good practice.