Suppose universal gravitational constant starts to decrease, then

Options

A. length of the day on the earth will increase
B. length of the year will increase
C. the earth will follow a spiral path of decreasing radius
D. kinetic energy of the earth will decrease

Explanation

To solve the problem regarding the effects of a decreasing universal gravitational constant (G) on various aspects of celestial mechanics, we can follow these steps:

### Step 1: Understand the relationship between gravitational force and centripetal force
The gravitational force between two masses (Earth and Sun) is given by the formula:
\[ F_g = \frac{G \cdot M_{Earth} \cdot M_{Sun}}{r^2} \]
where \( G \) is the universal gravitational constant, \( M_{Earth} \) and \( M_{Sun} \) are the masses of the Earth and Sun respectively, and \( r \) is the distance between them.

### Step 2: Relate gravitational force to centripetal force
For an object in circular motion (like the Earth around the Sun), the centripetal force required to keep the object in orbit is provided by the gravitational force:
\[ F_c = M_{Earth} \cdot a_c \]
where \( a_c \) is the centripetal acceleration, given by:
\[ a_c = \frac{v^2}{r} \]
Setting the gravitational force equal to the centripetal force gives us:
\[ \frac{G \cdot M_{Earth} \cdot M_{Sun}}{r^2} = M_{Earth} \cdot \frac{v^2}{r} \]

### Step 3: Simplify the equation
By canceling \( M_{Earth} \) from both sides and rearranging, we find:
\[ v^2 = \frac{G \cdot M_{Sun}}{r} \]
This shows that the orbital speed \( v \) depends on \( G \).

### Step 4: Analyze the effect of decreasing G on kinetic energy
The kinetic energy (KE) of the Earth in its orbit can be expressed as:
\[ KE = \frac{1}{2} M_{Earth} v^2 \]
Substituting for \( v^2 \):
\[ KE = \frac{1}{2} M_{Earth} \cdot \frac{G \cdot M_{Sun}}{r} \]
If \( G \) decreases, then the kinetic energy \( KE \) also decreases.

### Step 5: Determine the effect on the orbital period
The orbital period \( T \) can be derived from Kepler's third law:
\[ T^2 \propto \frac{r^3}{G \cdot M_{Sun}} \]
If \( G \) decreases, \( T^2 \) increases, which means \( T \) (the orbital period) increases. Thus, the length of the year increases.

### Step 6: Conclusion
Based on the analysis, if the universal gravitational constant \( G \) decreases:
- The kinetic energy of the Earth in its orbit decreases.
- The length of the year (orbital period) increases.

### Final Answer
The correct options are:
- The kinetic energy of the Earth decreases.
- The length of the year increases.