Chapter 8, Problem 9P

(a)

To determine

The angular velocity of the Earth in its orbit around the Sun.

Answer to Problem 9P

The angular velocity of the Earth in its orbit around the Sun is $\text{ 2}\times {\text{10}}^{-7}\text{ rad/sec}$

Explanation of Solution

Given:

Earth moves around the Sun by 360 degrees, so $\Delta \theta \text{ = 2π radians}$

Time taken for the rotation is 365 days.

Formula used:

The angular velocity of the Earth in its orbit around the Sun is determined by the following formula,

  $\text{Angular Velocity (ω) = }\frac{\text{Δθ }}{\text{Δt}}$

Calculation:

Substitute the given values,

  $\begin{array}{c}\text{ω = }\frac{\text{2π radians }}{\text{365 days}}\\ =\frac{\text{2π radians }}{\text{365}\times \text{24}\times \text{60}\times \text{60 seconds}}\\ =\frac{\text{2π radians }}{\text{3}\text{.15}\times {\text{10}}^{\text{7}}\text{seconds}}\\ =2.0\times {10}^{-7}\text{ rad/sec}\end{array}$

Conclusion:

The angular velocity of the Earth in its orbit around the Sun is $\text{2}\times {\text{10}}^{-7}\text{ rad/sec}$

(b)

To determine

Angular velocity of Earth about its axis.

Answer to Problem 9P

Angular velocity of Earth about its axis is $\text{7}\text{.3}\times {\text{10}}^{-5}\text{ rad/sec}$

Explanation of Solution

Given:

Earth moves about its axis by 360 degrees, so $\Delta \theta \text{ = 2π radians}$

Time taken for this rotation is 1 day.

Formula used:

The angular velocity of the Earth about its axis is determined by the following formula,

  $\text{Angular Velocity (ω) = }\frac{\text{Δθ }}{\text{Δt}}$

Calculation:

Substitute the given values,

  $\begin{array}{c}\text{ω = }\frac{\text{2π radians }}{\text{1 day}}\\ =\frac{\text{2π radians }}{\text{1}\times \text{24}\times \text{60}\times \text{60 seconds}}\\ =\frac{\text{2π radians }}{\text{8}\text{.64}\times {\text{10}}^4\text{seconds}}\\ =7.3\times {10}^{-5}\text{ rad/sec}\end{array}$

Conclusion:

Angular velocity of the Earth about its axis is $\text{ 7}\text{.3}\times {\text{10}}^{-5}\text{ rad/sec}$