As the earth rotates, what is the speed of (a) a physics student in Miami, Florida, at latitude 26 degrees

Verified step by step guidance

1. Determine the radius of the Earth at the latitude of Miami, Florida. The formula to find the radius at a given latitude \( \theta \) is \( R_{\text{latitude}} = R \cos(\theta) \), where \( R \) is the Earth's average radius (approximately 6371 km) and \( \theta \) is the latitude in degrees.
2. Convert the latitude of Miami from degrees to radians because trigonometric functions in calculators typically use radians. Use the conversion formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
3. Calculate the circumference of the Earth at the latitude of Miami using the formula \( C = 2\pi R_{\text{latitude}} \), where \( R_{\text{latitude}} \) is the radius calculated in step 1.
4. Determine the rotational period of the Earth, which is approximately 24 hours. Convert this period into seconds to match the units typically used in speed calculations.
5. Calculate the speed of the physics student in Miami by dividing the circumference from step 3 by the time period in seconds from step 4. The formula for speed is \( v = \frac{\text{distance}}{\text{time}} \).

Key Concepts

Rotational Speed

Rotational speed refers to how fast an object rotates around an axis. For 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 its Effect on Speed

Latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. As latitude increases from the equator to the poles, the circumference of the Earth decreases, resulting in a lower linear speed for objects at higher latitudes. For example, a student in Miami, located at 26 degrees latitude, will have a different rotational speed compared to someone at the equator.

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 this force is not directly responsible for the speed calculation, it is important to understand the dynamics of motion on a rotating body like Earth.