A fast-moving superhero in a comic book runs around a circular, 70-m-diameter track five and a half times (ending up directly opposite her starting point) in 3.0s.
What is her angular speed, in rad/s?
Va=5.5rev/3.0s * 6.28rad/rev=11.5 rad/s
To find the angular speed of the superhero, we can use the formula:
Angular Speed = Total Angle Covered / Time Taken
First, let's calculate the total angle covered by the superhero. The superhero runs around the 70 m diameter track 5.5 times, which means she covers a distance equal to the circumference of the track 5.5 times.
Circumference = π × diameter
Circumference = π × 70 m
Circumference = 220 m
Total Distance Covered = 220 m × 5.5 = 1210 m
Now we need to calculate the total angle covered. Since the superhero ends up directly opposite her starting point, she has completed a full circle. Therefore, the total angle covered is 2π radians.
Total Angle Covered = 2π radians
Now we can substitute the values into the formula:
Angular Speed = Total Angle Covered / Time Taken
Angular Speed = 2π radians / 3.0 s
Now let's calculate the result:
Angular Speed = (2 × 3.14159 radians) / 3.0 s
Angular Speed ≈ 2.094 radians/s
Therefore, the superhero's angular speed is approximately 2.094 radians per second.
To find the angular speed of the superhero, we need to calculate the total angle covered by her as she completes the circular track.
The circumference of a circle is given by the formula C = 2πr, where r is the radius. In this case, the diameter of the track is 70 m, so the radius (r) is half of that, which is 35 m.
The total distance covered in one complete loop around the track is equal to the circumference, so the distance covered in one loop is C = 2π(35) = 70π m.
Since the superhero runs around the track 5.5 times, the total distance covered is given by 5.5 times the distance covered in one loop, which is 5.5 * 70π m.
We are given the time it takes for the superhero to complete this distance, which is 3.0 seconds.
To find the angular speed, we need to divide the total angle covered by the time taken. The total angle is equal to the distance covered divided by the radius (r).
Total angle = Total distance / Radius
Total angle = (5.5 * 70π m) / 35 m
Simplifying this expression, we get:
Total angle = 11π radians
Angular speed = Total angle / Time
Angular speed = (11π radians) / (3.0 seconds)
Now, we can calculate the angular speed:
Angular speed ≈ 11π / 3 rad/s
Therefore, the angular speed of the superhero in the comic book is approximately 11π / 3 rad/s.