The earth rotates on its axis once every 24 hours.
How long does it take Earth to rotate through an angle of 4(pi)/3?
One rotation is 2π radians, so you can set up a ratio
(4π/3) / (2π) = x/24
x = 24(4π/3) / (2π)
= 16
It would take 16 hrs
or
look at it this way:
4π/3 radians = 240°
so it does 240/360 or 2/3 of a rotation
2/3 of 24 hours = 16 hours
1 rotation is 24 hours which is equal to 2pi.
W = 2pi/24
W = pi/12
W = pi/t
Pi/12 = 4pi/3 divided by t
t = 4pi/3 * 12/pi
t = 4/3 * 12
t = 16
It’s bigger than black and white, it’s a problem with the whole way of life. Can’t change over night, but we gotta start somewhere-
Well, if the Earth rotates 360 degrees in 24 hours, we can use some math to figure out how long it takes to rotate through an angle of 4(pi)/3.
Let's break it down:
360 degrees = 2(pi) radians
So, 4(pi)/3 radians would be (4/3) * (2(pi)) = (8/3)(pi) radians.
To find out the time it takes to rotate through this angle, we can set up a proportion:
360 degrees / 24 hours = (8/3)(pi) radians / x hours
Now, let me grab my handy-dandy calculator... *beep boop beep*
After some quick calculations, using the power of math... I'd say it takes approximately... drumroll, please... 0.3492 hours, or roughly 20.95 minutes for the Earth to rotate through an angle of 4(pi)/3!
To find out how long it takes for Earth to rotate through an angle of 4π/3, we need to know the relationship between angles and time.
The Earth completes one full rotation when it rotates through an angle of 2π radians. So, we can set up a proportion to find the time it takes to rotate through any other angle.
Let's call the time it takes for Earth to rotate through an angle θ as T.
In one rotation, Earth turns through 2π radians, which takes 24 hours. So, we have the following proportion:
2π radians / 24 hours = θ radians / T hours
To find T, we can cross-multiply and solve for T:
2πT = 24θ
Dividing both sides by 2π, we get:
T = 12θ / π
Now, we can substitute the given angle θ = 4π/3 into this equation:
T = 12(4π/3) / π
Simplifying, we have:
T = 48π / 3π
Cancelling out π, we get:
T = 16 hours.
Therefore, it takes Earth approximately 16 hours to rotate through an angle of 4π/3.