how far from the centre of the earth do you need to go for "g" to be reduced to 0.5 of its value on the surface of Earth?
a) 0.41 rE
b) 0.50 rE
c) 0.71 rE
d) 1.4 rE
e) 2.0 rE
Please help me <3
Why did the student take a shovel to the library? Because they heard they needed to dig deep into the subject of gravity!
Alright, let me calculate that for you. According to my calculations, the correct option is:
c) 0.71 rE
So, if you go approximately 0.71 times the radius of the Earth away from the center, you'll experience a gravitational force that is 0.5 of its value on the surface of the Earth. Happy exploring!
To determine how far from the center of the Earth you need to go for the acceleration due to gravity (g) to be reduced to 0.5 of its value on the surface of the Earth, you can use the formula for gravitational acceleration:
g = (G * M) / r^2
Where:
- g is the acceleration due to gravity on the surface of the Earth
- G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the Earth (5.972 x 10^24 kg)
- r is the distance from the center of the Earth
We want to find the value of r when g is reduced to 0.5g, so we can set up the following equation:
0.5g = (G * M) / r^2
Now, we can rearrange the equation to solve for r:
r^2 = (G * M) / (0.5g)
r^2 = (2 * G * M) / g
r = sqrt((2 * G * M) / g)
Let's calculate the value of r using the given values:
g = acceleration due to gravity on the surface of the Earth = 9.8 m/s^2
G = gravitational constant = 6.67430 x 10^-11 m^3 kg^-1 s^-2
M = mass of the Earth = 5.972 x 10^24 kg
r = sqrt((2 * 6.67430 x 10^-11 * 5.972 x 10^24) / 9.8)
After calculations, r is approximately 0.41 times the radius of the Earth (rE). Therefore, the answer is option a) 0.41 rE.
To find the distance from the center of the Earth where the acceleration due to gravity (g) is reduced to 0.5 of its value on the Earth's surface, we can use the formula:
g' = (1 / (1 + x)) * g
where g' is the new value of gravity, x is the distance from the center of the Earth, and g is the value of gravity on Earth's surface.
Let's solve this equation for x:
0.5g = (1 / (1 + x)) * g
Divide both sides of the equation by g:
0.5 = 1 / (1 + x)
Cross-multiply:
0.5(1 + x) = 1
0.5 + 0.5x = 1
Subtract 0.5 from both sides:
0.5x = 0.5
Divide both sides by 0.5:
x = 1
Therefore, when x = 1, the acceleration due to gravity is reduced to 0.5 of its value on the Earth's surface. This means the distance from the center of the Earth is equal to the radius of the Earth (rE).
Among the provided options, the correct answer would be:
b) 0.50 rE
since F is inversely proportional to r^2, you need to increase r by a factor of √2
Then F is changed to F' = GMm/(√2r)^2 = GMM/(2r^2) = F/2