Change in Earth's Radius

To determine the change in the value of 'g' on Earth's surface when its radius shrinks by 4% while the mass remains the same, we need to understand the relationship between gravitational acceleration ('g'), mass ('M'), and radius ('R').

Gravitational Acceleration Formula

The formula for gravitational acceleration is given by:
g = G * (M / R^2)

Where:
g is the gravitational acceleration
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3⋅kg^-1⋅s^-2)
M is the mass of the Earth
R is the radius of the Earth

Effect of Shrinking Radius on 'g'

When the radius of the Earth shrinks by 4%, the new radius (R') can be calculated as:
R' = R - (0.04 * R) = 0.96 * R

The mass of the Earth remains the same, so M' = M.

Substituting the new radius and the unchanged mass into the gravitational acceleration formula, we get:
g' = G * (M' / R'^2)

Calculating the New Value of 'g'

Substituting the values, we have:
g' = G * (M / (0.96 * R)^2)

Expanding the equation and simplifying, we get:
g' = G * (M / (0.96^2 * R^2))
g' = G * (M / (0.9216 * R^2))
g' = (1 / 0.9216) * G * (M / R^2)
g' = 1.085 * G * (M / R^2)

Conclusion

The value of 'g' on Earth's surface will increase by approximately 8.5% when the radius shrinks by 4% while the mass remains the same.

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