It takes 23 hours 56 minutes and 4 seconds for the earth to make one revolution (mean sidereal day). What is the angular speed of the earth?
Assume the earth is spherical. Relative to someone on the rotation axis, what is the linear speed of an object on the surface if the radius vector from the center of the earth to the object makes an angle of 43.0
The Correct Answer and Explanation is :
To find the angular speed of the Earth, we can use the formula for angular speed ((\omega)):
[
\omega = \frac{\text{Total angle of rotation}}{\text{Time for one rotation}}
]
Since the Earth makes one full rotation (360 degrees or (2\pi) radians) in a sidereal day, which is 23 hours, 56 minutes, and 4 seconds, we first need to convert this time into seconds:
[
23 \text{ hours} \times 3600 \text{ seconds/hour} + 56 \text{ minutes} \times 60 \text{ seconds/minute} + 4 \text{ seconds} = 86164 \text{ seconds}
]
Now, we calculate (\omega):
[
\omega = \frac{2\pi \text{ radians}}{86164 \text{ seconds}} \approx 0.000072921 \text{ radians/second}
]
Next, to find the linear speed (v) of an object on the surface of the Earth at a latitude making an angle of (43^\circ) from the equator, we use the formula for linear speed:
[
v = R \omega \cos(\theta)
]
where (R) is the radius of the Earth (approximately 6371 km or 6,371,000 meters), (\omega) is the angular speed we calculated, and (\theta) is the latitude angle from the equator. Since the angle provided is from the rotation axis, we use (90^\circ – 43^\circ = 47^\circ). Thus:
[
v = 6,371,000 \times 0.000072921 \times \cos(47^\circ)
]
First, calculate (\cos(47^\circ)):
[
\cos(47^\circ) \approx 0.681998
]
Now, find (v):
[
v \approx 6,371,000 \times 0.000072921 \times 0.681998 \approx 313.04 \text{ meters/second}
]
The angular speed of the Earth is approximately 0.000072921 radians per second, indicating how fast the Earth rotates in radians per second. The linear speed at the surface for an object located at a latitude corresponding to 43 degrees from the rotation axis (47 degrees from the equator) is about 313 meters per second. This speed is a measure of how fast a point on the Earth’s surface moves due to the Earth’s rotation. This motion contributes to phenomena like the Coriolis effect, which affects weather patterns and ocean currents.
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