If the law of gravitational, instead of being inverse-square law, becomes an inverse-cube law

Options

A. planets will not have elliptic obits
B. circular orbits of planets is not possible
C. projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic
D. there will be no gravitational force inside a spherical shell of uniform density

Explanation

To solve the problem, we need to analyze the implications of changing the law of gravitation from an inverse-square law to an inverse-cube law. Let's break down the analysis step by step.

### Step 1: Understanding the Inverse-Cube Law
In the inverse-square law, the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
If we change this to an inverse-cube law, the force becomes:
\[ F = \frac{G m_1 m_2}{r^3} \]

### Step 2: Analyzing Orbital Motion
For a planet in orbit, the gravitational force provides the necessary centripetal force. The centripetal force required for circular motion is given by:
\[ F_c = \frac{m v^2}{r} \]
Setting the gravitational force equal to the centripetal force:
\[ \frac{G m_1 m_2}{r^3} = \frac{m v^2}{r} \]
This simplifies to:
\[ v^2 = \frac{G m_2}{r^2} \]
Now, using Kepler's third law, we know:
\[ T^2 \propto r^3 \]
However, with the inverse-cube law, we find:
\[ T^2 \propto r^4 \]
This indicates that planets will not have elliptical orbits since the relationship does not hold. Thus, the first option is **correct**.

### Step 3: Circular Orbits
The second option states that circular orbits are not possible. However, we see that if the gravitational force can still provide the necessary centripetal force, circular orbits can still exist. Therefore, this statement is **incorrect**.

### Step 4: Projectile Motion
The third option discusses projectile motion. The acceleration due to gravity \( g \) would change to:
\[ g' = \frac{G m_2}{r^3} \]
This implies that the acceleration is constant for a given distance from the center of the planet. Thus, the trajectory of a projectile thrown at an angle will still be parabolic. Therefore, this option is **correct**.

### Step 5: Gravitational Force Inside a Spherical Shell
The last option states that there will be no gravitational force inside a spherical shell of uniform density. According to the shell theorem, this is true for the inverse-square law, but with the inverse-cube law, the gravitational force would not be zero inside the shell. Thus, this statement is **incorrect**.

### Conclusion
Based on the analysis:
- The first option is **correct**: Planets will not have elliptical orbits.
- The second option is **incorrect**: Circular orbits are still possible.
- The third option is **correct**: Projectile motion will remain approximately parabolic.
- The fourth option is **incorrect**: There will be gravitational force inside a spherical shell.

### Final Answer
The correct options are **1 and 3**.

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