The radius and mass of earth are increased by 0.5%. Which of the following statement is false at the surface of the earth

The correct option is: g will increase

Let's analyze each parameter at the surface of the Earth step-by-step to determine which statement is false when both the mass ($M$) and radius ($R$) of the Earth are increased by 0.5%.

1. Acceleration due to gravity ($g$):
The acceleration due to gravity at the surface of the Earth is given by the formula:
$g=\frac{GM}{R^2}$
Taking the natural logarithm on both sides and differentiating to find the fractional change, we get:
$\frac{dg}{g}=\frac{dM}{M}-2\frac{dR}{R}$
Given that the mass and radius both increase by 0.5%:
$\frac{dM}{M}=+0.5\%$
$\frac{dR}{R}=+0.5\%$
Substituting these values into the fractional change equation:
$\frac{dg}{g}=0.5\%-2(0.5\%)=0.5\%-1.0\%=-0.5\%$
This negative sign indicates that the acceleration due to gravity $g$ actually decreases by 0.5%. Therefore, the statement "g will increase" is false.

2. Escape velocity ($v_e$):
The escape velocity from the surface of the Earth is given by:
$v_e=\sqrt{\frac{2GM}{R}}$
Since both $M$ and $R$ increase by the same percentage (0.5%), their ratio $\frac{M}{R}$ remains constant:
$\frac{M\text{'}}{R\text{'}}=\frac{1.005M}{1.005R}=\frac{M/R}{}$
Thus, the escape velocity remains unchanged, making this statement true.

3. Gravitational Potential Energy ($U$):
The gravitational potential energy of a body of mass $m$ on the Earth's surface is given by:
$U=-\frac{GMm}{R}$
Since the ratio $\frac{M}{R}$ remains constant, the potential energy also remains unchanged, making this statement true.

Consequently, the only false statement among the options is that "g will increase".