If the radius of the earth were to shrink by \(1 \%\) its mass remaining the same, the acceleration due to gravity on the earth's surface would (A) decrease by \(2 \%\) (B) remain Unchanged (C) increase by \(2 \%\) (D) increases by \(1 \%\)

Short Answer

Expert verified
If the radius of the Earth were to shrink by 1% while its mass remains the same, the acceleration due to gravity on the Earth's surface would increase by approximately 2.03%. The answer is closest to (C) increase by 2%.

Step by step solution

01

Relevant Formulas

In order to solve the problem, we will need the formula for gravitational acceleration, which is: \( g = \frac{GM}{r^2} \) Where: - \(g\) is the gravitational acceleration - \(G\) is the gravitational constant (approximately \(6.674 \times 10^{-11} m^3 kg^{-1} s^{-2}\)) - \(M\) is the mass of the body (Earth in our case) - \(r\) is the radius of the body
02

Initial Gravitational Acceleration

First, let's determine the initial gravitational acceleration before the radius shrinks. We will define the initial values as: - Radius: \(r\) - Gravitational Acceleration: \(g\) Now we can express the initial gravitational acceleration as: \( g = \frac{GM}{r^2} \)
03

Shrunk Radius and New Gravitational Acceleration

The exercise tells us that the radius of the Earth shrinks by 1%. Therefore, the new radius will be: \( r' = r - 0.01r = 0.99r \) Now let's find the new gravitational acceleration (\(g'\)) with the new radius: \(g' = \frac{GM}{(0.99r)^2} = \frac{GM}{0.9801r^2}\)
04

Comparing New and Old Gravitational Acceleration

Now we will compare the new gravitational acceleration (\(g'\)) to the old one (\(g\)), by taking the ratio of \(g'\) to \(g\): \(\frac{g'}{g} = \frac{GM}{0.9801r^2} \times \frac{r^2}{GM} = \frac{1}{0.9801} \approx 1.0203\) Which indicates that the gravitational acceleration increased. To find the percentage increase, we can subtract 1 and multiply by 100: \((1.0203 - 1) \times 100 = 2.03 \% \)
05

Conclusion

Therefore, if the radius of the Earth were to shrink by 1% while the mass remains the same, the acceleration due to gravity on the Earth's surface would increase by approximately 2.03%. The answer is closest to (C) increase by 2%.

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