The earth circles the sun once a year. How much work would have to be done on the earth to bring it to rest relative to the sun, (ignore the rotation of earth about - its own axis) Given that mass of the earth is `6 × 10^(24)` kg and distance between the sun and earth is `1.5 × 10^(8)` km-

Options

A. `2.7xx10^(30)J`
B. `2.7xx10^(31)J`
C. `-2xx10^(33)J`
D. `+2.7xx10^(33)J`

Explanation

To solve the problem of how much work would have to be done on the Earth to bring it to rest relative to the Sun, we can follow these steps:

### Step 1: Identify the Given Values
- Mass of the Earth, \( M = 6 \times 10^{24} \) kg
- Distance from the Earth to the Sun, \( r = 1.5 \times 10^{8} \) km = \( 1.5 \times 10^{11} \) m (since \( 1 \) km = \( 1000 \) m)

### Step 2: Calculate the Circumference of the Earth's Orbit
The circumference \( C \) of the orbit can be calculated using the formula:
\[ C = 2 \pi r \]
Substituting the value of \( r \):
\[ C = 2 \pi (1.5 \times 10^{11}) \approx 9.4248 \times 10^{11} \text{ m} \]

### Step 3: Determine the Time for One Revolution
The time taken for one complete revolution of the Earth around the Sun is approximately:
\[ T = 365.25 \text{ days} = 365.25 \times 24 \times 3600 \text{ seconds} \approx 3.156 \times 10^7 \text{ seconds} \]

### Step 4: Calculate the Orbital Velocity of the Earth
The orbital velocity \( V \) can be calculated using the formula:
\[ V = \frac{C}{T} \]
Substituting the values of \( C \) and \( T \):
\[ V = \frac{9.4248 \times 10^{11}}{3.156 \times 10^7} \approx 29.865 \text{ m/s} \]

### Step 5: Calculate the Kinetic Energy of the Earth
The kinetic energy \( K \) of the Earth can be calculated using the formula:
\[ K = \frac{1}{2} M V^2 \]
Substituting the values of \( M \) and \( V \):
\[ K = \frac{1}{2} (6 \times 10^{24}) (29.865)^2 \]
Calculating \( V^2 \):
\[ V^2 \approx 29.865^2 \approx 893.4 \text{ m}^2/\text{s}^2 \]
Now substituting back to find \( K \):
\[ K \approx \frac{1}{2} (6 \times 10^{24}) (893.4) \approx 2.67 \times 10^{27} \text{ joules} \]

### Step 6: Conclusion
The work done to bring the Earth to rest relative to the Sun is equal to the kinetic energy of the Earth:
\[ \text{Work done} = K \approx 2.67 \times 10^{27} \text{ joules} \]