What is (a) the angular speed and (b) the linear speed of a point on Earth's surface at latitude 30^∘ N. Take the radius of the Earth to be 6309 km. (c) At what latitude would your linear speed be 10 m / s ?

To calculate the angular speed � ω and linear speed � v of a point on Earth's surface at latitude 3 0 ∘ 30 ∘ North, we can use the following formulas: (a) Angular speed ( � ω): � = � � ω= r v ​ Where: � v is the linear speed. � r is the radius of the Earth. (b) Linear speed ( � v): � = � � v=rω Where: � ω is the angular speed. � r is the radius of the Earth. Given: Radius of the Earth, � = 6309 r=6309 km = 6309 × 1 0 3 6309×10 3 m Let's calculate: (a) Angular speed ( � ω): � = � � ω= r v ​ We know that Earth completes one full rotation in 24 hours, so the angular speed of Earth's rotation is 2 � 24 × 3600 24×3600 2π ​ radians per second. (b) Linear speed ( � v): � = � � v=rω (c) To find the latitude where the linear speed is 10 10 m/s, we can rearrange the formula for linear speed: � = � � v=rω � = � � ω= r v ​ Given the desired linear speed � = 10 v=10 m/s, we can find the corresponding latitude. Let's calculate these values:

Angular and Linear Speed on Earth's Surface

To calculate the angular speed \( \omega \) and linear speed \( v \) of a point on Earth's surface at latitude \( 30^\circ \) North:

(a) Angular Speed (\( \omega \)):

We know that Earth completes one full rotation in 24 hours, so the angular speed of Earth's rotation is given by:

\[ \omega = \frac{2\pi}{24 \times 3600} \, \text{rad/s} \]

(b) Linear Speed (\( v \)):

Using the formula \( v = r \omega \), we can calculate the linear speed:

\[ v = 6309 \times 10^3 \times \frac{2\pi}{24 \times 3600} \, \text{m/s} \]

(c) Latitude for Linear Speed of 10 m/s:

Given the linear speed \( v = 10 \) m/s, we can find the corresponding latitude using the formula \( \omega = \frac{v}{r} \).