Assertion : When the orbital radius of a planet is made 4 times, its time period increases by 8 times
Reason : Greater the height above the Earth's surface, greater is the time period of revolution

Options

A. Both A and are true and R is correct explanation of the assertion
B. Both A and R are true but R is not the correct explanation of the assertion
C. A is true, but R is false .
D. A is false, but R is true

Explanation

To solve the assertion and reason question, we need to analyze both the assertion and the reason using Kepler's third law of planetary motion. Here’s a step-by-step solution:

### Step 1: Understand the Assertion
The assertion states that when the orbital radius of a planet is made 4 times larger, its time period increases by 8 times.

### Step 2: Apply Kepler's Third Law
According to Kepler's third law, the square of the time period (T) of a planet is directly proportional to the cube of the orbital radius (R) of the planet:
\[ T^2 \propto R^3 \]
This can be expressed mathematically as:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

### Step 3: Set Up the Ratios
Let’s denote the initial radius as \( R_1 \) and the new radius as \( R_2 = 4R_1 \). Let \( T_1 \) be the initial time period and \( T_2 \) be the new time period. According to the assertion, we need to check if \( T_2 = 8T_1 \).

### Step 4: Substitute the Values
Substituting \( R_2 \) into the equation:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{(4R_1)^3} \]
This simplifies to:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{64R_1^3} = \frac{1}{64} \]

### Step 5: Solve for \( T_2^2 \)
Cross-multiplying gives:
\[ T_1^2 = \frac{1}{64} T_2^2 \]
Rearranging gives:
\[ T_2^2 = 64 T_1^2 \]
Taking the square root of both sides:
\[ T_2 = 8 T_1 \]
Thus, the assertion is correct.

### Step 6: Understand the Reason
The reason states that the greater the height above the Earth's surface, the greater the time period of revolution. This is also true because as the height increases, the orbital radius increases, leading to a longer time period.

### Step 7: Conclusion on the Reason
While both the assertion and reason are correct, the reason does not specifically explain the assertion. It does not mention Kepler's third law, which is the fundamental principle behind the relationship between orbital radius and time period.

### Final Answer
Both the assertion and reason are true, but the reason is not the correct explanation of the assertion.