Question

Your car's wheels are \(65 \mathrm{cm}\) in diameter and the wheels are spinning at an angular velocity of 101 rad/s. How fast is your car moving in kilometers per hour (assume no slippage)?

Short answer

Question: Given that the diameter of a car's wheels is 65 cm and their angular velocity is 101 rad/s, calculate the speed of the car in kilometers per hour. Answer: The car is moving at a speed of approximately 118.17 km/h.

Step-by-step solution

Write down the provided information

We are given the diameter of the wheels, \(65 \mathrm{cm}\), and the angular velocity of the wheels, \(101 \mathrm{rad/s}\). Our goal is to find the linear velocity of the car in kilometers per hour.

Calculate the radius of the wheels

To find the radius of the wheels, divide the diameter by 2: \(radius = \frac{diameter}{2} = \frac{65 \mathrm{cm}}{2} = 32.5 \mathrm{cm}\)

Convert the radius to meters

To convert the radius from centimeters to meters, divide by 100: \(radius_m = \frac{32.5 \mathrm{cm}}{100} = 0.325 \mathrm{m}\)

Relate angular and linear velocity

The formula relating angular velocity (\(\omega\)) to linear velocity (\(v\)) is: \(v = r\omega\) Plug in the values for the radius and angular velocity: \(v = (0.325 \mathrm{m})(101 \mathrm{rad/s}) = 32.825 \mathrm{m/s}\)

Convert meters per second to kilometers per hour

To convert meters per second to kilometers per hour, multiply by \(\frac{3600}{1,000}\): \(v_{km/h} = (32.825 \mathrm{m/s}) \cdot \frac{3600}{1000} = 118.17 \mathrm{km/h}\) So, the car is moving at a speed of approximately \(118.17 \mathrm{km/h}\).