The Slowing Earth The Earth's rate of rotation is constantly decreasing, causing the day to increase in duration. In the year 2006 the Earth took about 0.840 longer to complete 365 revolutions than it did in the year 1906.

What was the average angular acceleration of the Earth during this time? Give your answer in rad/s2

The average angular acceleration of the Earth, α = 6.152 X 10⁻²⁰ rad/s²

Explanation:

Given data,

The period of 365 rotation of Earth in 2006, T₁ = 365 days, 0.840 sec

= 3.1536 x 10⁷ + 0.840

= 31536000.84 s

The period of 365 rotation of Earth in 2006, T₀ = 365 days

= 31536000 s

Therefore, time period of one rotation on 2006, Tₐ = 31536000.84/365

= 86400.0023 s

The time period of rotation is given by the formula,

Tₐ = 2π / ωₐ

ωₐ = 2π / Tₐ

Substituting the values,

ωₐ = 2π / 365.046306

= 7.272205023 x 10⁻⁵ rad/s

Therefore, the time period of one rotation on 1906, Tₓ = 31536000/365

= 86400 s

Time period of rotation,

Tₓ = 2π / ωₓ

ωₓ = 2π / T

= 2π / 86400

= 7.272205217 x 10⁻⁵ rad/s

The average angular acceleration

α = (ωₓ - ωₐ) / T₁

= (7.272205217 x 10⁻⁵ - 7.272205023 x 10⁻⁵) / 31536000.84

α = 6.152 X 10⁻²⁰ rad/s²

Hence the average angular acceleration of the Earth, α = 6.152 X 10⁻²⁰ rad/s²