The Earth's rate of rotation is constantly decreasing, causing the day to increase in duration. In the year 2006 the Earth took about .840 s longer to complete 365 revolutions than it did in the year 1906. What was the average acceleration of the Earth during this time? Give your answer in rad/s^2
Well, I'm not sure about the exact average acceleration, but I can tell you the Earth must have been feeling a little sluggish during that time. It's like the Earth was saying, "Hey, I need some more beauty sleep, okay?" So maybe the Earth just wanted to take its time and enjoy each revolution a little more. After all, it's not easy spinning around all the time. It's a tough job, but someone's gotta do it!
To determine the average acceleration of the Earth during this time period, we need to calculate the change in angular velocity and the time for that change to occur.
First, let's calculate the change in angular velocity:
Change in angular velocity (Δω) = Change in time (Δt) / Number of revolutions (Δθ)
The change in time is given as 0.840 seconds longer, and the number of revolutions is 365. Thus:
Δt = 0.840 s
Δθ = 365 revolutions
Now, we need to convert the number of revolutions to radians:
1 revolution = 2π radians
Δθ (in radians) = 365 revolutions * 2π radians/revolution
Next, we can calculate the change in angular velocity:
Δω = Δt / Δθ
Substituting the given values:
Δω = 0.840 s / (365 * 2π radians/revolution)
Now, we can calculate the average acceleration using the formula:
Average acceleration (α) = Δω / Δt
Substituting the values:
α = (0.840 s / (365 * 2π radians/revolution)) / 0.840 s
Simplifying:
α = 1 / (365 * 2π radians/revolution)
Finally, let's calculate the numerical value of the average acceleration:
α ≈ 1.7321 x 10^(-7) rad/s^2
Therefore, the average acceleration of the Earth during this time period is approximately 1.7321 x 10^(-7) rad/s^2.
To find the average acceleration of the Earth during the given time period, we need to calculate the change in angular velocity and divide it by the change in time.
First, let's calculate the change in time:
The difference between the years 2006 and 1906 is 100 years. Since there are 365 days in a year, the total number of days between these two years is 100 * 365 = 36,500 days.
To convert this to seconds, we multiply by 24 (hours/day) * 60 (minutes/hour) * 60 (seconds/minute) = 31,536,000 seconds.
Now, let's calculate the change in angular velocity:
The difference in the number of revolutions is given as 365 revolutions. To convert this to radians, we multiply by 2π since one revolution is equal to 2π radians.
Change in angular velocity (Δω) = 365 * 2π radians = 730π radians.
Finally, we can calculate the average acceleration:
Average acceleration (α) = Δω / Δt.
α = 730π radians / 31,536,000 seconds.
Calculating this value, we get:
α ≈ 0.00007332 rad/s^2.
Therefore, the average acceleration of the Earth during this time period is approximately 0.00007332 rad/s^2.
Average angular accleration =
(change in angular velocity)/(100 years)
Angular velocity = 2 pi radians/24 hours
= 7.276*10^-5
relative change in angular velocity =
(0.840 s/86400 s) = 9.72*10^-6
absolute change in angular velocity
-7.07*10^-10 rad/s
Now go back to the first formula, and divide by the number of seconds in 100 years.