Suppose the Earth was only half the size it is now (half the mass and half the radius), what would "g" be?
Newtons law:
g= G Me/r^2
Now, if one halves the radius, mass is not half.
Mass=density*volume=density 4/3 PI r^3 so if radius goes down by 1/2, then mass goes to 1/8 Me
So your question is really confusing.
Newton's law of universal gravitation:
where force equals the universal constant of gravity multiplied by mass of the earth and mass of an object divided by the raduis squared
F = G((m1*m2)/(r^2))
Force equals the mass of an object multiplied by the acceleration or 'g'
mg = G((m1*m2)/(r^2))
Mass of the object would cancel out on both sides since infact we do not have a second mass
g = G((m1)/(r^2))
The universal constant of gravity as Newton discovered is:
G = 6.67 * 10^-11 m^3/kg*s^2
The mass of the Earth is:
mass of the earth = 5.98 * 10^24 kg
The radius of the Earth is:
radius of the earth = 6.38 * 10^6 m
The mass then needs to be divided in half:
5.98 * 10^24 kg/2 = 2.94 * 10^24 kg
The radius then needs to be divided in half:
6.38 * 10^6 m/2 = 3.19 * 10^6 m
The equation for the acceleration or 'g' is:
g = 6.67 * 10^-11 m^3/kg*s^2((2.94 * 10^24 kg)/(3.19 * 10^6 m^2))
Leaving 'g' as:
g = 19.27 m/s^2
To determine the value of "g" (acceleration due to gravity) on a hypothetical Earth that is half the size of the current Earth (half the mass and half the radius), we can use the formula for gravitational acceleration:
g = G * (M / R^2)
Where:
g = acceleration due to gravity
G = gravitational constant
M = mass of the Earth
R = radius of the Earth
In this scenario, we need to calculate the values of "M" and "R" for the smaller Earth.
Given:
M (mass of current Earth) = M1
R (radius of current Earth) = R1
The mass of the smaller Earth (M2) would be half of the current Earth:
M2 = M1 / 2
The radius of the smaller Earth (R2) would also be half of the current Earth:
R2 = R1 / 2
Now we can substitute these values into the formula:
g2 = G * (M2 / R2^2)
Since M2 = M1 / 2 and R2 = R1 / 2:
g2 = G * ((M1/2) / (R1/2)^2)
To simplify the equation, we can rewrite R1/2 as (1/2)^2 = 1/4:
g2 = G * ((M1/2) / (1/4))
Next, we can multiply the numerator by 4 and simplify:
g2 = G * (4 * M1 / 2)
g2 = G * (2 * M1)
Therefore, the value of "g" on a hypothetical Earth that is half the size of the current Earth would be twice the current value.