As the earth rotates, what is the speed of a physics student in Miami, Florida, at latitude 26°. Ignore the revolution of the earth around the sun. The radius of the earth is 6400 km.

검증된 단계별 안내

Determine the radius of the Earth at the equator, which is approximately 6378 km. Convert this value to meters for consistency in SI units: \( R_{\text{Earth}} = 6.378 \times 10^6 \ \text{m} \).

Calculate the effective radius of the circular path of the student in Miami. Since the student is at a latitude of 26 degrees, the effective radius is given by \( R_{\text{effective}} = R_{\text{Earth}} \cdot \cos(\theta) \), where \( \theta = 26^{\circ} \).

Determine the angular velocity of the Earth's rotation. The Earth completes one full rotation in 24 hours, so the angular velocity is \( \omega = \frac{2\pi}{T} \), where \( T = 24 \ \text{hours} = 24 \cdot 3600 \ \text{seconds} \).

Use the formula for linear speed in circular motion: \( v = R_{\text{effective}} \cdot \omega \). Substitute the values of \( R_{\text{effective}} \) and \( \omega \) into this equation to find the speed of the student.

Simplify the expression to calculate the final speed. Ensure all units are consistent (meters and seconds) throughout the calculation.

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주요 개념

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Rotational Speed

Rotational speed refers to how fast an object rotates around an axis. For the Earth, this speed varies with latitude; points on the equator move faster than those near the poles due to the Earth's spherical shape. The rotational speed can be calculated using the formula v = ωr, where v is linear speed, ω is angular velocity, and r is the radius of the circular path.

Latitude and Circumference

Latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. It affects the circumference of the circular path an object follows due to Earth's rotation. At latitude 26 degrees, the effective radius of rotation is smaller than at the equator, leading to a lower linear speed for a person located there.

Centrifugal Force

Centrifugal force is an apparent force that acts outward on a body moving around a center, arising from the body's inertia. As the Earth rotates, this force affects objects on its surface, including a physics student in Miami. While it does not directly influence the speed calculation, it is important for understanding the effects of Earth's rotation on weight and motion.