The linear velocity of a point on the surface of earth at a latitude of `60^@` is

Options

A. `800/3ms^(-1)`
B. `(800pi)/3ms^(-1)`
C. `800xx5/18ms^(-1)`
D. `(2000pi)/27ms^(-1)`

Answer

To find the linear velocity of a point on the surface of the Earth at a latitude of \(60^\circ\), we can follow these steps:

### Step 1: Understand the relationship between angular velocity and linear velocity
The linear velocity \(V\) of a point on the surface of the Earth can be calculated using the formula:
\[ V = \omega r \]
where:
- \(V\) is the linear velocity,
- \(\omega\) is the angular velocity in radians per second,
- \(r\) is the radius of the horizontal circle at the given latitude.

### Step 2: Calculate the angular velocity \(\omega\)
The Earth completes one rotation (360 degrees or \(2\pi\) radians) in 24 hours. Therefore, the angular velocity \(\omega\) can be calculated as:
\[ \omega = \frac{2\pi \text{ radians}}{T} \]
where \(T\) is the time period in seconds. Since \(T = 24 \text{ hours} = 24 \times 60 \times 60 \text{ seconds}\):
\[ T = 86400 \text{ seconds} \]
Thus,
\[ \omega = \frac{2\pi}{86400} \text{ radians/second} \]

### Step 3: Calculate the radius \(r\) at latitude \(60^\circ\)
The radius of the Earth \(R\) is approximately \(6400 \text{ km} = 6400 \times 1000 \text{ m}\). The radius \(r\) of the horizontal circle at latitude \(\phi\) is given by:
\[ r = R \cos(\phi) \]
For \(\phi = 60^\circ\):
\[ r = 6400 \times 1000 \times \cos(60^\circ) \]
Since \(\cos(60^\circ) = \frac{1}{2}\):
\[ r = 6400 \times 1000 \times \frac{1}{2} = 3200 \times 1000 \text{ m} = 3,200,000 \text{ m} \]

### Step 4: Calculate the linear velocity \(V\)
Now substituting \(\omega\) and \(r\) into the linear velocity formula:
\[ V = \omega r = \left(\frac{2\pi}{86400}\right) \times (3200 \times 1000) \]
Calculating this gives:
\[ V = \frac{2\pi \times 3200 \times 1000}{86400} \]
\[ V = \frac{6400\pi}{86400} = \frac{2\pi}{27} \text{ m/s} \]

### Final Result
The linear velocity of a point on the surface of the Earth at a latitude of \(60^\circ\) is:
\[ V = \frac{2000\pi}{27} \text{ m/s} \]