Suppose the law of gravitational attraction suddenly changes and becomes an inverse cube law i.e. `F prop 1//r^3` , but still remaining a central force. Then

Options

A. Kepler’s law of areas still holds
B. Kepler’s law of period still holds
C. Kepler’s law of period still holds
D. Neither the law of areas, nor the law of period still holds

Explanation

To solve the problem, we need to analyze the implications of a gravitational force that follows an inverse cube law, \( F \propto \frac{1}{r^3} \), while still being a central force. We will examine how this change affects Kepler's laws of planetary motion.

### Step-by-Step Solution:

1. **Understanding the Force Law**:
The original law of gravitation states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) is given by:
\[ F = G \frac{m_1 m_2}{r^2} \]
If the law changes to an inverse cube law, it becomes:
\[ F = k \frac{m_1 m_2}{r^3} \]
where \( k \) is a constant.

2. **Analyzing Kepler's First Law**:
Kepler's First Law states that planets move in elliptical orbits with the Sun at one focus. The shape of the orbit is determined by the nature of the gravitational force. For the inverse square law, the orbits are conic sections (ellipses, parabolas, or hyperbolas). However, with an inverse cube law, the force does not provide the necessary centripetal force to maintain elliptical orbits. Therefore, Kepler's First Law does not hold.

3. **Analyzing Kepler's Second Law**:
Kepler's Second Law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This law is derived from the conservation of angular momentum. For an inverse cube law, the angular momentum is not conserved in the same way as it is for an inverse square law. Thus, Kepler's Second Law also does not hold.

4. **Analyzing Kepler's Third Law**:
Kepler's Third Law relates the square of the orbital period \( T \) of a planet to the cube of the semi-major axis \( a \) of its orbit:
\[ T^2 \propto a^3 \]
This relationship is derived from the gravitational force being proportional to \( \frac{1}{r^2} \). With an inverse cube law, the relationship between the period and the semi-major axis changes, and hence, Kepler's Third Law does not hold.

5. **Conclusion**:
Since all three of Kepler's laws are derived based on the inverse square law of gravitation, they do not hold true when the law changes to an inverse cube law. Therefore, the correct conclusion is that neither Kepler's laws of area nor period hold in this new scenario.

### Final Answer:
None of Kepler's laws hold true under the new inverse cube law of gravitation.