Latitude is an angle that ranges from 0° at Earth's equator to 90° (north or south) at the poles. What is the angular speed w and linear speed of a point along Earth's surface at 30° north latitude?
To find the angular speed (w) at 30° north latitude, we first need to know the Earth's angular speed. The Earth takes approximately 24 hours to complete a full rotation of 360° around its axis. Therefore:
w_earth = 360° / (24 hours) = 15° per hour
Since the angular speed is the same everywhere on Earth, the angular speed (w) at 30° north latitude is also:
w = 15° per hour
Next, we need to find the linear speed (v) at 30° north latitude. To do this, we first calculate the radius (r) of the circle at 30° north latitude using Earth's radius (R):
R = 6371 km (approximate Earth's radius)
r = R × cos(latitude)
r = 6371 km × cos(30°)
r ≈ 5520.571 km
Now we can calculate the circumference (C) of the circle at 30° north latitude:
C = 2 × π × r
C ≈ 2 × π × 5520.571 km
C ≈ 34,669.991 km
Since the Earth takes 24 hours to complete a full rotation (360°), the linear speed at 30° north latitude is:
v = C / (24 hours)
v ≈ 34,669.991 km / 24 hours
v ≈ 1444.583 km/h
Thus, the angular speed (w) at 30° north latitude is 15° per hour, and the linear speed (v) is approximately 1444.583 km/h.
To calculate the angular speed and linear speed of a point along Earth's surface at a specific latitude, we first need to know the rotational speed of the Earth. The Earth completes a full rotation in about 24 hours, so its rotational speed can be calculated as follows:
Rotational speed (ω) = 360° / 24 hours = 15° per hour
Next, we need to take into consideration the latitude at which we want to calculate the angular speed and linear speed. In this case, we are interested in 30° north latitude.
At the equator (0° latitude), the linear speed of a point along Earth's surface is at its maximum, while it decreases as we move towards higher latitudes. The linear speed (v) can be calculated using the following formula:
v = ω * r * cos(latitude)
Where:
ω = rotational speed (in radians per hour)
r = radius of the Earth
latitude = the desired latitude in degrees
Let's calculate the angular speed and linear speed at 30° north latitude:
Angular speed (w):
w = ω * cos(latitude)
w = (15° per hour) * cos(30°)
w ≈ 15° per hour * 0.866 ≈ 13.0° per hour
Linear speed (v):
We need to convert the rotational speed from degrees per hour to radians per hour:
ω = (15° per hour) * (2π radians/360°) = (π/12) radians per hour
Using the Earth's radius (r ≈ 6,371 km):
v = ω * r * cos(latitude)
v = (π/12) * 6,371 km * cos(30°)
v ≈ 1,676 km/hour
Therefore, at 30° north latitude, the angular speed is approximately 13.0° per hour, and the linear speed is approximately 1,676 km/hour.
To find the angular speed, we need to determine the rate at which Earth rotates. Earth completes one rotation in approximately 24 hours, or 1440 minutes.
The total distance covered by the point along Earth's surface in one rotation is equal to the Earth's circumference, which is approximately 40,075 kilometers.
At 30° north latitude, the point is located at a distance of approximately 1/3 from the equator to the North Pole. This means it covers 1/3 of the Earth's circumference in one rotation.
Angular speed, denoted by w, is defined as the angle covered per unit of time. In this case, it is the angle covered by the point at 30° latitude in one minute.
To find the angular speed:
w = (1/3 * 360°) / 1440 minutes
w = (120°) / 1440
w ≈ 0.0833° per minute
Now, to find the linear speed of the point, we can use the relationship between angular speed and linear speed.
Linear speed (v) is calculated by multiplying the angular speed (w) by the radius (r) of the circular path.
At 30° north latitude, the radius of Earth's rotation (r) is equal to the distance from the Earth's center to the point at 30° latitude.
To calculate the radius:
r = radius of Earth * cosine(latitude)
r = 6,371 km * cos(30°)
r ≈ 5,511 km
To find the linear speed:
v = w * r
v ≈ 0.0833° per minute * 5,511 km
v ≈ 460 km per minute
Therefore, the angular speed is approximately 0.0833° per minute, and the linear speed of a point at 30° north latitude is approximately 460 kilometers per minute.