What is the ratio of the wight of a man when he is in a tunnel 1000 km below the surface of the Earth to his weight at the surface?
...I'm not sure how to proceed, not because I don't know how to do it, but because I have no idea how to come up with a(g) 1000 km inside the earth. I tried applying the law of shells (or whatever it's called) and:
a(g) = [GM(ins)]/(R^2)
*M(ins) = p*(4/3)*pi*R^3
so:
a(g) = [G * p*(4/3)*pi*R^3]/(R^2)
= 3 G M/(4 * pi * R^2)
...but when I end up plugging that back in, I get a value of a(g) that is like 3.0, which doesn't make any sense since we're closer to the center of the earth, so shouldn't a(g) be larger??
When calculating acceleration due to gravity inside the Earth, we need to consider two important facts:
1. As we go deeper into the Earth, the mass of the Earth inside the radius at that depth becomes responsible for gravity, not the whole mass of the Earth.
2. According to the Shell Theorem, when we are inside a spherical shell, the net gravitational force experienced by a point mass is zero.
So, when we are 1000 km below the surface, the mass of the Earth above us (in a shell with a thickness of 1000 km) has no effect on our weight. We only need to consider the mass of the Earth inside the sphere with radius 6371 - 1000 = 5371 km.
Let's denote the total mass of the Earth as M_e and radius as R_e. The mass of Earth inside the smaller radius, R_ins (5371 km), can be called M_ins. We can find M_ins by assuming Earth has uniform density:
Density of Earth = p_e = M_e / ((4/3) * pi * R_e ^ 3)
Now, we can find M_ins with this density and the sphere of radius R_ins:
M_ins = p_e * (4/3) * pi * R_ins ^ 3
Now, we can find acceleration due to gravity at 1000 km below the surface:
a(g)_ins = (GM_ins) / (R_ins ^ 2)
Dividing a(g)_ins by a(g)_surface (or g), we get the ratio:
Ratio = a(g)_ins / g = [(GM_ins) / (R_ins ^ 2)] / [(GM_e) / (R_e ^ 2)] = (M_ins * R_e ^ 2) / (M_e * R_ins ^ 2)
Plugging in the values of M_ins, R_e, and R_ins in the ratio, we can find the actual ratio.
The ratio you are calculating above is slightly off, since you mixed some terms. But if you follow the steps I explained, you should get a correct answer. And the gravity inside the Earth should be less as we go deeper.
The weight of an object at any point inside the Earth can be calculated using the formula:
W = mg
where W is the weight of the object, m is its mass, and g is the acceleration due to gravity at that point.
To determine the ratio of the weight of the man when he is 1000 km below the surface of the Earth to his weight at the surface, we need to compare the acceleration due to gravity at those two locations.
At the surface of the Earth, the acceleration due to gravity can be approximated as 9.8 m/s^2. However, as you correctly pointed out, the value of g changes as we move deeper into the Earth due to the distribution of mass.
To calculate the acceleration due to gravity, a(g), 1000 km below the surface of the Earth, we need to utilize the formula:
a(g) = GM(r)/(r^2)
where G is the gravitational constant, M(r) is the mass enclosed within a radius r, and r is the distance from the center of the Earth.
You attempted to use the formula for a general shell to calculate a(g), which was a good approach. However, your misunderstanding lies in the calculation of the mass enclosed within that radius.
The mass enclosed within a radius r can be expressed as:
M(r) = (4/3)*pi*rho*(r^3)
where rho is the average density of the Earth.
So, to calculate the ratio of the weight of the man when he is 1000 km below the surface of the Earth to his weight at the surface, we can express it as:
W(1000 km below) / W(surface) = [m*g(1000 km below)] / [m*g(surface)]
Now let's calculate the values step-by-step:
Step 1: Calculate g(1000 km below)
- Use the formula a(g) = GM(r)/(r^2), where r = radius of the Earth - 1000 km.
Step 2: Calculate g(surface)
- Use the approximate value of the acceleration due to gravity at the surface, which is 9.8 m/s^2.
Step 3: Find the ratio
- Divide g(1000 km below) by g(surface) to get the ratio.
Please let me know if you would like to proceed with these steps and calculate the ratio.
To determine the weight of a man when he is 1000 km below the surface of the Earth compared to his weight at the surface, you can use the concept of gravitational acceleration.
First, let's break down the steps to calculate the weight ratio:
1. Calculate the gravitational acceleration at the surface of the Earth, denoted by "g".
- The value for g is approximately 9.8 m/s^2.
2. Calculate the radius of the Earth, denoted by "R".
- The average radius of the Earth is approximately 6,371 km.
3. Calculate the distance from the surface to the desired location inside the Earth. In this case, it's 1000 km (or 1,000,000 meters).
4. Use Newton's Law of Universal Gravitation to calculate the gravitational acceleration at the specified depth, denoted by "g_inside".
- The formula is: g_inside = (G * M_inside) / (R_inside^2)
- Here, G is the gravitational constant (approximately 6.67430 × 10^(-11) N m^2/kg^2).
- M_inside represents the mass inside the radius at the specified depth.
- R_inside is the radius of the sphere at the specified depth.
After calculating g_inside, you can find the weight ratio by dividing g_inside by g.
To address your concern about the value of g_inside being smaller, it is important to note that as you move deeper into the Earth, the mass above you starts to cancel out, resulting in a net decrease in gravitational acceleration. This effect causes the gravitational acceleration at the specified depth to be lower than at the surface. Therefore, the weight ratio will be less than 1.
Hope this explanation helps clarify the process of calculating the weight ratio!