Question

Take the radius of the Earth to be 6309 km

1)The angular speed of a point on Earth’s surface at latitude 30° N is ____ x 10^-4 rad/s

2)The linear speed of a point on Earth’s surface at latitude 30° N is

3)The latitude where your linear speed is 10 m/s is ±

Answer 1

1) The angular speed at this point is the same as the angular speed of the Earth:

$$\omega_{Earth}=\dfrac{2\pi\ rad}{0.997\ d\cdot\dfrac{24\ h}{1\ d}\cdot\dfrac{3600\ s}{1\ h}}=7.29\cdot10^{-5}\ \dfrac{rad}{s}.$$

2) We can find the linear speed of a point on Earth’s surface at latitude 30° N as follows:

$$v=\omega_{Earth}R_{Earth}cos\theta,$$ $$v=7.29\cdot10^{-5}\ \dfrac{rad}{s}\cdot6.37\cdot10^6\ m\cdot cos30^{\circ}=402\ \dfrac{m}{s}.$$

3) We can find the latitude as follows:

$$v=\omega_{Earth}R_{Earth}cos\theta,$$ $$\theta=cos^{-1}\ (\dfrac{v}{\omega_{Earth}R_{Earth}}),$$ $$\theta=cos^{-1}\ (\dfrac{10\ \dfrac{m}{s}}{7.29\cdot10^{-5}\ \dfrac{rad}{s}\cdot6.37\cdot10^6\ m})=\pm88.76^{\circ}.$$