If both the mass and radius of the earth decrease by `1%` the value of

Options

A. acceleration due to gravity would decrease by nearly `1%`
B. acceleration due to gravity would increase by `1%`
C. escape velocity from the earth's surface would decrease by `1%`
D. the gravitational potential energy of a body on earth's surface will remain unchanged

Explanation

To solve the problem step by step, we will analyze how the changes in mass and radius of the Earth affect the acceleration due to gravity (g), escape velocity, and gravitational potential energy.

### Step 1: Understand the formula for acceleration due to gravity
The acceleration due to gravity (g) at the surface of the Earth is given by the formula:
\[ g = \frac{G \cdot M}{R^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Calculate the new values of mass and radius
If both the mass and radius of the Earth decrease by 1%, we can express the new values as:
- New mass \( M' = 0.99M \)
- New radius \( R' = 0.99R \)

### Step 3: Substitute the new values into the formula for g
Substituting the new values into the formula for g, we get:
\[ g' = \frac{G \cdot M'}{(R')^2} = \frac{G \cdot (0.99M)}{(0.99R)^2} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ g' = \frac{G \cdot (0.99M)}{(0.9801R^2)} \]
This can be rewritten as:
\[ g' = \frac{0.99}{0.9801} \cdot \frac{G \cdot M}{R^2} \]
\[ g' = \frac{0.99}{0.9801} \cdot g \]

### Step 5: Calculate the ratio
Calculating the ratio:
\[ \frac{0.99}{0.9801} \approx 1.01 \]
Thus, we find:
\[ g' \approx 1.01g \]
This means that the new value of \( g \) has increased by approximately 1%.

### Step 6: Analyze escape velocity
The escape velocity \( v_e \) is given by:
\[ v_e = \sqrt{2gR} \]
Substituting the new values:
\[ v_e' = \sqrt{2g'R'} = \sqrt{2 \cdot (1.01g) \cdot (0.99R)} \]
This simplifies to:
\[ v_e' = \sqrt{2gR} \cdot \sqrt{1.01 \cdot 0.99} \]
Since \( \sqrt{1.01 \cdot 0.99} \) is approximately 1, the escape velocity remains nearly unchanged.

### Step 7: Analyze gravitational potential energy
The gravitational potential energy \( U \) is given by:
\[ U = -\frac{G \cdot M \cdot m}{R} \]
Substituting the new values:
\[ U' = -\frac{G \cdot (0.99M) \cdot m}{0.99R} \]
This simplifies to:
\[ U' = -\frac{G \cdot M \cdot m}{R} \]
Thus, the gravitational potential energy remains unchanged.

### Conclusion
From the analysis:
- The acceleration due to gravity increases by approximately 1%.
- The escape velocity remains unchanged.
- The gravitational potential energy remains unchanged.

### Final Answer
The correct options are:
- Option 2: Acceleration due to gravity would increase by one percent.
- Option 4: Gravitational potential energy of a body on Earth's surface will remain unchanged.