The radius of the earth's very nearly circular orbit around the sun is 1.5 x 10^11m. Find the magnitude of the earth's (b) angular velocity

Verified step by step guidance

1. Identify the formula for angular velocity, which is \(\omega = \frac{v}{r}\), where \(v\) is the linear velocity and \(r\) is the radius of the circular path.
2. Recognize that the Earth completes one full orbit around the Sun in one year. Use this information to find the linear velocity \(v\) by calculating the circumference of the orbit (\(2\pi r\)) and dividing it by the orbital period (one year, converted to seconds).
3. Substitute the value of \(r\) (1.5 x 10^11 m) into the circumference formula to find \(v\).
4. Calculate the angular velocity \(\omega\) by dividing the linear velocity \(v\) by the radius \(r\).
5. Express the angular velocity in radians per second, ensuring the units are consistent throughout the calculations.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point or axis. It is defined as the rate of change of angular displacement and is typically expressed in radians per second. For circular motion, it can be calculated using the formula ω = θ/t, where θ is the angle in radians and t is the time taken. In the context of Earth's orbit, it describes how fast the Earth moves around the Sun.

Centripetal Motion

Centripetal motion refers to the motion of an object that is moving in a circular path, requiring a net inward force directed towards the center of the circle. This force is essential for maintaining circular motion and is provided by gravitational attraction in the case of planetary orbits. The centripetal acceleration can be calculated using the formula a_c = v^2/r, where v is the linear velocity and r is the radius of the circular path.

Orbital Period

The orbital period is the time it takes for an object to complete one full orbit around another object. For Earth, this period is approximately one year, which is the time it takes to travel around the Sun. The relationship between the orbital period and angular velocity is given by the formula ω = 2π/T, where T is the orbital period. Understanding this relationship is crucial for calculating the angular velocity of Earth in its orbit.