If radius of earth decreases by 10%, then acceleration due to gravity (Density of earth remain constant)

Options

A. Decreases by 10%
B. Decreases by 19%
C. Increases by 21%
D. Increases by 10%

Answer

To solve the problem of how the acceleration due to gravity changes when the radius of the Earth decreases by 10%, while keeping the density constant, we can follow these steps:

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

### Step 2: Relate mass and density
Since the density \( \rho \) of the Earth remains constant, we can express the mass \( M \) in terms of density and volume:
\[ M = \rho \cdot V = \rho \cdot \left(\frac{4}{3} \pi R^3\right) \]
Substituting this into the formula for \( g \):
\[ g = \frac{G \cdot \rho \cdot \left(\frac{4}{3} \pi R^3\right)}{R^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho R \]
This shows that \( g \) is directly proportional to \( R \) when density is constant.

### Step 3: Calculate the new radius after a 10% decrease
If the radius \( R \) decreases by 10%, the new radius \( R_f \) can be calculated as:
\[ R_f = R - 0.1R = 0.9R \]

### Step 4: Find the new acceleration due to gravity
Using the proportionality we established, the new acceleration due to gravity \( g_f \) can be expressed as:
\[ g_f = \frac{4}{3} \pi G \rho R_f = \frac{4}{3} \pi G \rho (0.9R) = 0.9 \left(\frac{4}{3} \pi G \rho R\right) = 0.9g \]

### Step 5: Calculate the percentage change in acceleration due to gravity
The percentage change in \( g \) can be calculated as:
\[ \text{Percentage change} = \frac{g_f - g}{g} \times 100\% \]
Substituting \( g_f = 0.9g \):
\[ \text{Percentage change} = \frac{0.9g - g}{g} \times 100\% = \frac{-0.1g}{g} \times 100\% = -10\% \]

### Conclusion
Thus, the acceleration due to gravity decreases by 10%.

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