Problem 19 - College Physics

Question

a car is traveling at a constant speed on the highway. Its tires have a diameter of \(61.0 \mathrm{~cm}\) and are rolling without sliding or slipping. If the angular speed of the tires is \(50.0 \mathrm{rad} / \mathrm{s},\) what is the speed of the car, in SI units?

Step-by-Step Solution

Answer

The speed of the car is 15.25 m/s.

Step 1: Understand the Problem

We need to find the linear speed of the car based on the angular speed of its tires. We know the diameter of the tires and the angular speed.

Step 2: Convert Diameter to Radius

The radius of the tire is needed to calculate the linear speed. The diameter is given as 61.0 cm, so the radius is half of that, which is: \( r = \frac{61.0}{2} \) cm. Convert this to meters: \( r = \frac{61.0}{2} \times \frac{1}{100} \ m = 0.305 \ m \).

Step 3: Use the Relationship Between Linear and Angular Speed

The formula to relate linear speed \( v \) of the car to angular speed \( \omega \) of the tire is: \( v = r \times \omega \). We can now substitute \( r = 0.305 \ m \) and \( \omega = 50.0 \ rad/s \) into this formula to find \( v \).

Step 4: Calculate the Linear Speed

Substitute the values into the formula: \( v = 0.305 \ m \times 50.0 \ rad/s \). This gives \( v = 15.25 \ m/s \).

Step 5: Express the Result in SI Units

The speed we've calculated is in meters per second (\( m/s \)), which is the SI unit for speed. Therefore, the speed of the car is 15.25 \( m/s \).

Key Concepts

Angular Speed

When we talk about angular speed, we're referring to how fast an object rotates around an axis. It's important in understanding the motion of wheels, propellers, and even planets! Angular speed is measured in radians per second

• One complete revolution is equal to 360 degrees, or 2π radians.
• Angular speed, denoted as \( \omega \), indicates how many radians an object covers per second.

For example, if a tire rotates at an angular speed of 50 radians per second, it means it turns through 50 rad in one second. This concept is key to relating how fast something is spinning to how fast it is moving in a straight line.

Rolling Without Slipping

When a wheel rolls without slipping, its motion is smooth and the point of contact with the ground doesn't slide. This is essential in keeping the speed and distance traveled accurate.

• Each point on the edge of the tire travels a distance equal to the tire's circumference in one full rotation.
• The friction between the tire and the road ensures the tire rolls without skidding.

The relationship between angular speed and linear speed ("v") is only valid when there's no slipping. In our car example, the smooth motion of rolling allows us to calculate linear speed using angular data. When applying this concept:\[ v = r \times \omega \]we assume the tire rolls perfectly without slipping.

Radius Calculation

To find linear speed from angular speed, you need the radius of the wheel. This is because the radius connects the rotational aspect (angular speed) with the linear aspect (linear speed).

• To convert a diameter to radius: divide the diameter by 2.
• Conversion to meters is crucial as we work in SI units, where 1 meter equals 100 centimeters.

With a tire diameter of 61 cm, the radius is calculated as\[ r = \frac{61}{2} = 30.5 \text{ cm} \]or\[ r = 0.305 \text{ m} \]This is the value used in the linear speed formula, ensuring we compute speeds in coherent SI units.

SI Units

SI Units, or the International System of Units, are the standard for scientific measurements. They ensure consistency and comparability across various scientific fields by using a common base.

• The SI unit for length is the meter (m), and for time, it is the second (s).
• Speed is the distance covered per unit time, with meters per second (m/s) as its official SI unit.

In this problem, we initially have measurements in centimeters, but converting to meters (the SI unit for length) ensures we can calculate the car's speed properly. Working in SI units allows for seamless applications in physics, engineering, and other disciplines.