Problem 9 - College Physics

Question

A car with a 65-cm-diameter wheel travels \(3.0 \mathrm{~km}\). How many revolutions does the wheel make in this distance?

Step-by-Step Solution

Answer

The wheel makes approximately 1469 revolutions.

Step 1: Determine the Wheel's Circumference

The circumference of a circle is given by the formula \( C = \pi \cdot d \), where \( d \) is the diameter. For a 65 cm-diameter wheel, the circumference is: \[ C = \pi \times 65 \text{ cm} = 65\pi \text{ cm}. \]

Step 2: Convert Distance Traveled to Centimeters

The total distance traveled by the car is 3.0 km. We convert this distance to centimeters because the wheel's circumference is in centimeters. \[ 3.0 \text{ km} = 3.0 \times 1000 \text{ m} = 3000 \text{ m} \] and \[ 3000 \text{ m} = 3000 \times 100 \text{ cm} = 300000 \text{ cm}. \]

Step 3: Calculate the Number of Revolutions

To find the number of revolutions, divide the total distance traveled by the wheel's circumference: \[ \text{Number of Revolutions} = \frac{300000 \text{ cm}}{65\pi \text{ cm}}. \] Simplifying this gives approximately: \[ \frac{300000}{65 \times 3.14159} \approx 1468.57. \]

Step 4: Round to the Nearest Whole Number

The number of revolutions should be a whole number because a wheel can't partially revolve. Therefore, the number of revolutions is approximately 1469.

Key Concepts

Circular Motion

Circular motion is when an object moves along a circular path. Think about a Ferris wheel or a car driving around a circular track. In our exercise, the car's wheel rotates in a circular motion as the car travels forward. Each complete spin of the wheel represents one revolution.

• A revolution is one complete turn around a circle, bringing it back to its starting point.
• This concept helps us figure out how many times the wheel spins to cover a certain distance.

Understanding circular motion is crucial in physics and engineering since many systems involve rotation and cycles.

Unit Conversion

In physics, understanding and correctly converting units is essential. Here we converted kilometers to centimeters since the wheel's measurements were in centimeters.

• 1 kilometer equals 1000 meters, and 1 meter equals 100 centimeters.
• Thus, 3.0 kilometers becomes 300,000 centimeters.

Making sure all measurements are in the same units allows us to easily calculate and avoid errors. Consistency in units ensures our calculated answers are accurate and make sense.

Mathematical Calculation

Solving physics problems often involves precise mathematical calculations. In our exercise, we needed to divide the total distance by the circumference of the wheel to find out how many times the wheel turned.

• We used the formula to find the number of revolutions: \( \text{Number of Revolutions} = \frac{\text{Total Distance}}{\text{Wheel's Circumference}} \).
• This calculation gives us an idea of how many circles the wheel rotated while the car moved 3 kilometers.

Precision is key; hence, converting numbers into approximate decimals ensures we have a realistic count of revolutions.

Geometry in Physics

Geometry is an essential part of physics. It helps us understand shapes, sizes, and how they relate to different physical problems. In this case, the wheel's shape is a circle and its circumference helps us relate to its rotations.

• The circumference determines the edge length of a circle.
• It helps track how far the wheel travels with each rotation.
• This relationship shows how geometry supports physical calculations and understanding.

By grasping geometric concepts, we can tackle real-world problems, like determining the number of wheel revolutions a car makes, accurately and effectively.