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 3.0 s .
What is her angular speed, in rad/s?
d = 5.5rev * 6.28rad/rev = 34.6 radians.
V = 34.6rad./3s. = 11.5 rad/s.
To find the angular speed of the superhero, we first need to determine the total angle covered by the superhero as she runs around the circular track.
The formula to calculate the angle is:
θ = (2π * N) / n
Where:
- θ represents the angle covered by the superhero
- N is the number of times the superhero goes around the track (5.5 times)
- n is the total number of laps required to cover the entire circular track
Given that the superhero goes around the track five and a half times, we can say that N = 5.5. Since the superhero ends up directly opposite her starting point, she covers a total of 2n laps.
Using this information, we can calculate:
θ = (2π * 5.5) / (2 * 70)
Simplifying:
θ = (11π) / 70
Now, we need to determine the time taken by the superhero to cover the angle θ. Given that the superhero takes 3.0 seconds to complete 5.5 laps, we can write:
t = (3.0 * 2 * 70) / (5.5 * 2 * π)
Simplifying:
t = (6 * 70) / (11π)
Now, we can find the angular speed, ω (omega), which is given by the formula:
ω = θ / t
Plugging in the values:
ω = [(11π) / 70] / [(6 * 70) / (11π)]
Simplifying:
ω = (11π)^2 / (6 * 70^2)
Calculating this expression will give us the angular speed in rad/s.