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![**Problem 1**
A truck is moving at a constant speed. The diameter of its tires is 101.0 cm. If the angular speed of the tires is 54.0 rad/s, what is the (linear) speed of the truck? (Tires are rolling without slipping or sliding).
\[ v = ? \]
---
**Solution Explanation:**
To solve for the linear speed \( v \) of the truck, we will use the relationship between linear speed, angular speed, and the radius of the tires:
\[ v = \omega \times r \]
where \( \omega \) is the angular speed and \( r \) is the radius of the tire.
First, convert the diameter of the tire to the radius:
\[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{101.0 \, \text{cm}}{2} = 50.5 \, \text{cm} \]
Convert centimeters to meters since standard SI units should be used:
\[ r = 50.5 \, \text{cm} \times \frac{1 \, \text{m}}{100 \, \text{cm}} = 0.505 \, \text{m} \]
Now, apply the angular speed (\( \omega \)):
\[ \omega = 54.0 \, \text{rad/s} \]
Finally, substitute \( \omega \) and \( r \) into the formula:
\[ v = \omega \times r = 54.0 \, \text{rad/s} \times 0.505 \, \text{m} = 27.27 \, \text{m/s} \]
So, the linear speed \( v \) of the truck is:
\[ v = 27.27 \, \text{m/s} \]](https://content.bartleby.com/qna-images/question/a49c042c-e715-46d7-b1af-568fb3890c2a/2cfbedbd-0b13-4f26-9893-374b36d20ab6/atkgqnm_processed.jpeg)
Transcribed Image Text:**Problem 1**
A truck is moving at a constant speed. The diameter of its tires is 101.0 cm. If the angular speed of the tires is 54.0 rad/s, what is the (linear) speed of the truck? (Tires are rolling without slipping or sliding).
\[ v = ? \]
---
**Solution Explanation:**
To solve for the linear speed \( v \) of the truck, we will use the relationship between linear speed, angular speed, and the radius of the tires:
\[ v = \omega \times r \]
where \( \omega \) is the angular speed and \( r \) is the radius of the tire.
First, convert the diameter of the tire to the radius:
\[ \text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{101.0 \, \text{cm}}{2} = 50.5 \, \text{cm} \]
Convert centimeters to meters since standard SI units should be used:
\[ r = 50.5 \, \text{cm} \times \frac{1 \, \text{m}}{100 \, \text{cm}} = 0.505 \, \text{m} \]
Now, apply the angular speed (\( \omega \)):
\[ \omega = 54.0 \, \text{rad/s} \]
Finally, substitute \( \omega \) and \( r \) into the formula:
\[ v = \omega \times r = 54.0 \, \text{rad/s} \times 0.505 \, \text{m} = 27.27 \, \text{m/s} \]
So, the linear speed \( v \) of the truck is:
\[ v = 27.27 \, \text{m/s} \]
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