The latitude of Nashville TN is 47.0 degree N. What is its rotational speed, assuming the radius of the earth at the equator to be 6380 km?
A 0.339 m/s
B 316 m/s
C 0.498 m/s
D 339 m/s
E 498 m/s
F 0.316 m/s
The Correct Answer and Explanation is :
To determine the rotational speed of Nashville, Tennessee, given its latitude, we need to use the Earth’s rotation and geometry. The problem provides:
- Latitude of Nashville: ( 47.0^\circ ) N
- Radius of Earth at the equator: ( 6380 \, \text{km} )
- We want to calculate the rotational speed (tangential speed) at this latitude.
Step 1: Formula for rotational speed
The rotational speed ( v ) at a given latitude can be determined by:
[
v = \frac{2 \pi R \cos(\theta)}{T}
]
Where:
- ( R ) is the radius of the Earth at the equator (6380 km, or ( 6.38 \times 10^6 \, \text{m} )),
- ( \theta ) is the latitude (47.0°),
- ( T ) is the time for one complete rotation of the Earth (24 hours, or 86400 seconds).
Step 2: Effective radius at latitude
At a latitude of ( 47.0^\circ ), the effective radius of Earth decreases by a factor of ( \cos(\theta) ). So the effective radius at Nashville is:
[
R_{\text{effective}} = R \cos(\theta)
]
[
R_{\text{effective}} = 6.38 \times 10^6 \, \text{m} \times \cos(47^\circ)
]
[
R_{\text{effective}} = 6.38 \times 10^6 \, \text{m} \times 0.682
]
[
R_{\text{effective}} = 4.35 \times 10^6 \, \text{m}
]
Step 3: Calculate the rotational speed
Now, we can substitute this effective radius into the formula for ( v ):
[
v = \frac{2 \pi \times 4.35 \times 10^6}{86400}
]
[
v \approx \frac{2 \pi \times 4.35 \times 10^6}{86400} \approx 316.1 \, \text{m/s}
]
Step 4: Answer
The rotational speed at Nashville is approximately 316 m/s. Therefore, the correct answer is B: 316 m/s.
Explanation
The Earth rotates once every 24 hours. The closer you are to the equator, the faster you move due to the larger radius of the Earth. As you move toward the poles (like Nashville, at 47° N), your speed decreases because the radius of the circular path you follow becomes smaller. We calculated the effective radius for Nashville and then used the formula for rotational speed to find the value of approximately 316 m/s.