Answer :
Answer:
To calculate the angular velocity (\(\omega\)) and linear velocity (\(v\)) of the Earth in its orbit around the Sun, we can use the following formulas:
1. Angular velocity (\(\omega\)) is given by:
\[ \omega = \frac{2 \pi}{T} \]
where \(T\) is the period of the Earth's orbit around the Sun.
2. Linear velocity (\(v\)) is given by:
\[ v = r \times \omega \]
where \(r\) is the radius of the Earth's orbit around the Sun.
Given:
Radius of Earth's orbit (\(r\)) = \(1.5 \times 10^{11}\) m
First, we need to find the period of the Earth's orbit (\(T\)). The period of Earth's orbit around the Sun is approximately one year, which is \(T = 365\) days or \(T = 365 \times 24 \times 3600\) seconds.
Let's calculate:
\[ T = 365 \times 24 \times 3600 \]
\[ T = 31,536,000 \text{ seconds} \]
Now, we can calculate the angular velocity (\(\omega\)):
\[ \omega = \frac{2 \pi}{T} \]
\[ \omega = \frac{2 \times 3.14}{31,536,000} \]
\[ \omega ≈ 1.99 \times 10^{-7} \text{ rad/s} \]
Next, we can calculate the linear velocity (\(v\)):
\[ v = r \times \omega \]
\[ v = (1.5 \times 10^{11}) \times (1.99 \times 10^{-7}) \]
\[ v ≈ 29,850 \text{ m/s} \]
So, the angular velocity of the Earth in its orbit around the Sun is approximately \(1.99 \times 10^{-7}\) rad/s, and the linear velocity is approximately \(29,850\) m/s.