Which of the following statements is/are correct? 1. Angular velocity for all locations on the Earth's surface is the same while linear velocity varies. 2. Linear velocity is maximum at the equator and minimum at the poles. Select the correct answer using the code given below.
Both 1 and 2
Analyzing Earth's Rotational Velocity Statements
This question asks us to evaluate two statements concerning the motion of points on the Earth's surface due to its rotation. We need to determine the correctness of these statements about angular velocity and linear velocity at different locations on Earth.
Statement 1: Angular Velocity vs. Linear Velocity
The first statement claims that the angular velocity for all locations on the Earth's surface is the same, while the linear velocity varies.
- Angular Velocity (\({\omega}\)): The Earth completes one full rotation ( \(2\pi\) radians) in approximately 24 hours (the period, $T$). Angular velocity is defined as the angle swept per unit time. The formula is \({\omega = \frac{2\pi}{T}}\). Since the period $T$ is the same for every point on Earth (it takes the entire planet the same amount of time to complete one rotation), the angular velocity \({\omega}\) is constant for all points on the Earth's surface, regardless of their latitude.
- Linear Velocity ($v$): Linear velocity, also known as tangential velocity, is related to angular velocity by the formula \(v = r\omega\), where $r$ is the perpendicular distance of the point from the axis of rotation.
- Variation of $r$: For points on the Earth's surface, the distance $r$ from the axis of rotation is not constant. It is maximum at the equator (equal to the Earth's radius, $R$) and decreases as we move towards the poles, becoming zero at the poles themselves.
- Conclusion for Statement 1: Because $r$ varies, and \({\omega}\) is constant, the linear velocity $v$ (\(v = r\omega\)) must also vary for different points on the Earth's surface. Thus, Statement 1 is correct.
Statement 2: Velocity Variation Across Earth's Surface
The second statement asserts that linear velocity is maximum at the equator and minimum at the poles.
- As established above, the linear velocity is given by \(v = r\omega\).
- The angular velocity (\({\omega}\)) is constant everywhere on Earth.
- The distance from the axis of rotation ($r$) is maximum at the equator, where \(r = R\) (Earth's radius). The linear velocity at the equator is \(v_{eq} = R\omega\).
- The distance from the axis of rotation ($r$) is minimum (zero) at the poles, where \(r = 0\). The linear velocity at the poles is \(v_{pole} = 0 \times \omega = 0\).
- Conclusion for Statement 2: Since the linear velocity is directly proportional to the distance $r$ from the axis of rotation, it is indeed maximum at the equator (largest $r$) and minimum (zero) at the poles (smallest $r$). Thus, Statement 2 is also correct.
Final Conclusion on Earth's Motion
Based on the analysis of both statements:
- Statement 1 is correct because all points on Earth share the same angular velocity (\({\omega}\)), but their linear velocities ($v$) differ due to varying distances ($r$) from the rotation axis.
- Statement 2 is correct because the linear velocity (\(v = r\omega\)) is greatest where the distance from the axis ($r$) is greatest (the equator) and least where $r$ is smallest (the poles).
Therefore, both statements are correct.
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