Which of the following graphs between the square of the time period and cube of the distance of the planet from the sun is correct ?

Options

A.
B.
C.
D.

Answer

To solve the question regarding the relationship between the square of the time period (T²) and the cube of the distance (r³) of a planet from the sun, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Kepler's Third Law**:
According to Kepler's Third Law of planetary motion, the square of the time period (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit. This can be mathematically expressed as:
\[ T^2 \propto r^3 \]

2. **Express the Proportionality as an Equation**:
We can convert the proportionality into an equation by introducing a constant of proportionality (k):
\[ T^2 = k \cdot r^3 \]

3. **Identify the Variables**:
Here, we can identify:
- \( y = T^2 \) (the square of the time period)
- \( x = r^3 \) (the cube of the distance from the sun)

4. **Graphical Representation**:
Since \( T^2 \) is directly proportional to \( r^3 \), we can say:
\[ y = k \cdot x \]
This indicates that if we plot \( y \) (T²) on the y-axis and \( x \) (r³) on the x-axis, the graph will be a straight line passing through the origin.

5. **Conclusion**:
The correct graph between the square of the time period and the cube of the distance of the planet from the sun will be a straight line inclined to the x-axis. Therefore, the correct option is the one that represents this linear relationship.