Answer :
To determine the distance traveled by a car tire with a diameter of 60 centimeters turning with an angular velocity of 6 radians per second over a period of 5 minutes, we can follow these steps:
1. Convert the time from minutes to seconds:
- Time in minutes: [tex]\( 5 \)[/tex] minutes
- Time in seconds: [tex]\( 5 \text{ minutes} \times 60 \text{ seconds per minute} = 300 \text{ seconds} \)[/tex]
2. Calculate the radius of the tire:
- Diameter of the tire: [tex]\( 60 \)[/tex] centimeters
- Radius of the tire: [tex]\( \frac{\text{Diameter}}{2} = \frac{60 \text{ cm}}{2} = 30 \text{ cm} \)[/tex]
3. Calculate the total angular displacement (θ) in radians:
- Angular velocity: [tex]\( 6 \)[/tex] radians per second
- Time in seconds: [tex]\( 300 \)[/tex] seconds
- Total angular displacement: [tex]\( \theta = \text{Angular velocity} \times \text{Time} = 6 \text{ radians/second} \times 300 \text{ seconds} = 1800 \text{ radians} \)[/tex]
4. Determine the distance traveled by the tire:
- The distance traveled by a point on the perimeter of the tire is equal to the total angular displacement multiplied by the radius of the tire.
- Distance traveled: [tex]\( \text{Distance} = \theta \times \text{radius} = 1800 \text{ radians} \times 30 \text{ cm} = 54000 \text{ cm} \)[/tex]
Finally, to convert the distance from centimeters to meters (for a more standard unit of measure):
- [tex]\( 54000 \text{ cm} = 54000 \div 100 = 540 \text{ meters} \)[/tex]
Conclusion:
The distance traveled by the car tire in 5 minutes is [tex]\( 540 \)[/tex] meters.
1. Convert the time from minutes to seconds:
- Time in minutes: [tex]\( 5 \)[/tex] minutes
- Time in seconds: [tex]\( 5 \text{ minutes} \times 60 \text{ seconds per minute} = 300 \text{ seconds} \)[/tex]
2. Calculate the radius of the tire:
- Diameter of the tire: [tex]\( 60 \)[/tex] centimeters
- Radius of the tire: [tex]\( \frac{\text{Diameter}}{2} = \frac{60 \text{ cm}}{2} = 30 \text{ cm} \)[/tex]
3. Calculate the total angular displacement (θ) in radians:
- Angular velocity: [tex]\( 6 \)[/tex] radians per second
- Time in seconds: [tex]\( 300 \)[/tex] seconds
- Total angular displacement: [tex]\( \theta = \text{Angular velocity} \times \text{Time} = 6 \text{ radians/second} \times 300 \text{ seconds} = 1800 \text{ radians} \)[/tex]
4. Determine the distance traveled by the tire:
- The distance traveled by a point on the perimeter of the tire is equal to the total angular displacement multiplied by the radius of the tire.
- Distance traveled: [tex]\( \text{Distance} = \theta \times \text{radius} = 1800 \text{ radians} \times 30 \text{ cm} = 54000 \text{ cm} \)[/tex]
Finally, to convert the distance from centimeters to meters (for a more standard unit of measure):
- [tex]\( 54000 \text{ cm} = 54000 \div 100 = 540 \text{ meters} \)[/tex]
Conclusion:
The distance traveled by the car tire in 5 minutes is [tex]\( 540 \)[/tex] meters.
