# How does the gravitational acceleration on a surface of planet change, if its radius shrinks while retaining the same mass? [closed]

If the radius of the earth were to shrink by $$1$$% while its mass remained the same,

then how would the acceleration due to gravity on the earth's surface increase by $$2$$%?

## closed as off-topic by Thomas Fritsch, Jon Custer, John Rennie, G. Smith, YashasMay 31 at 16:25

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The initial acceleration of a mass on the Earth's surface would have the value $$a_i = \frac{GM}{r^2}.$$ Where $$M$$ is the mass of the Earth and $$r$$ the distance from the Earth's center. If the Earth shrinks by $$1 \%$$ then the final acceleration will be $$a_f = \frac{GM}{(0.99r)^2}.$$ The increase in acceleration will be equal to $$a_f/a_i = (0.99)^{-2} \simeq 1.02$$. Which is an increase of approximately $$2 \%$$.
The equation is: $$F=\frac{Gm_1m_2}{r^2}$$
Where F is the gravitational force, $$m_1$$ and $$m_2$$ are the masses of the two gravitationally interacting objects, $$r$$ is the distance between their centre of masses, and $$G$$ is the gravitational constant, equal to about $$6.67*10^{-11}$$.