The angular speed of a point on a planet is [tex]\frac{3 x }{10}[/tex] radian per hour. The equator lies on a circle of radius approximately 4000 miles. Find the linear velocity, in miles per hour, of a point on the equator.
The linear speed of a point on the equator is approximately $\square$ miles per hour
(Round to the nearest whole number as needed.)
Answer (2)
The angular speed is given as Ī = 10 3 x â radians per hour.
The radius of the equator is given as r = 4000 miles.
The linear velocity is calculated using the formula v = r Ī .
Substituting the given values, we find the linear velocity to be 1200 x â .
Explanation
Problem Analysis and Given Data The problem provides the angular speed of a point on a planet, Ī = 10 3 x â radians per hour, and the radius of the equator, r = 4000 miles. We need to find the linear velocity v of a point on the equator. The formula relating linear velocity, angular speed, and radius is v = r Ī .
Calculate Linear Velocity Substitute the given values into the formula: v = r Ī = 4000 Ã 10 3 x â Simplify the expression: v = 10 4000 Ã 3 x â = 10 12000 x â = 1200 x The linear velocity is 1200 x miles per hour.
Final Answer The linear speed of a point on the equator is approximately 1200 x miles per hour.
Examples
Understanding linear and angular velocities is crucial in many real-world applications. For instance, when designing gears or pulleys, engineers need to calculate the linear speed of a belt or chain based on the angular speed of the rotating components. Similarly, in astronomy, knowing the angular speed of a planet's rotation and its radius allows us to determine the linear speed of a point on its equator, which helps in understanding weather patterns and climate dynamics.
The linear velocity of a point on the equator can be calculated using the formula v = rĪ. By substituting the given radius and angular speed, we find that the linear velocity is 1200x miles per hour. This result illustrates how angular speed and radius combine to affect linear motion on circular paths.
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