To determine the new period of revolution for a satellite orbiting Earth after the planet's radius shrinks to half, we can use Kepler's third law of planetary motion, which relates the period of an orbiting body to the radius of its orbit. The law states that the square of the period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit. In mathematical terms, this is expressed as:
Understanding the Relationship
The formula can be simplified for circular orbits as follows:
T² ∝ r³
From this, we can derive the relationship:
T = k * r^(3/2)
where k is a constant that depends on the mass of the central body (in this case, Earth). Since the mass of Earth remains unchanged, we can focus on how the radius affects the period.
Initial Conditions
Let’s denote the initial radius of Earth as R and the initial period of the satellite as T. According to Kepler's law:
T² = k * R³
New Conditions After Radius Shrinkage
When the radius of Earth shrinks to half, the new radius becomes:
r = R/2
Now, we can substitute this new radius into the equation:
T'² = k * (R/2)³
This simplifies to:
T'² = k * (R³ / 8)
Since we know that T² = k * R³, we can substitute k * R³ with T²:
T'² = T² / 8
Finding the New Period
To find T', we take the square root of both sides:
T' = T / √8
Since √8 can be simplified to 2√2, we have:
T' = T / (2√2)
Now, we can express this in terms of the original period T:
T' = T / (2√2) = T * (1 / (2√2))
Final Calculation
To find the new period in relation to the original period T, we can express it as:
T' = T / (2√2) = T * (1 / 2.828) ≈ 0.353T
This indicates that the new period is less than the original period, but we need to express it in the options provided. The closest option that matches our derived period is:
None of the options directly match our calculation. However, if we consider the implications of the radius change, we can conclude that the period decreases significantly. The correct answer based on the derived relationship is not explicitly listed among the options, but it is important to recognize that the period will be shorter than T.
In summary, the new period of revolution for the satellite, after the radius of Earth shrinks to half, will be significantly less than the original period T, but the exact numerical representation does not match the provided options. This highlights the importance of understanding the underlying physics rather than just matching answers.
