Confusion on product rule for mass of differential volume element

AI Thread Summary
The discussion centers on the confusion regarding the product rule in calculating the mass of a planet with a non-uniform density that decreases linearly from the center to the surface. The initial approach mistakenly equated density as a function of radius to mass over volume, leading to incorrect results after differentiation. It was clarified that the correct relationship for mass should involve integrating density over volume, rather than using the average density at a specific radius. The key takeaway is that the mass should be expressed as an integral of density, avoiding the erroneous assumption that density can be treated as constant during differentiation. This highlights the importance of accurately representing the relationship between density and mass in varying density scenarios.
TRB8985
Messages
74
Reaction score
15
Homework Statement
Planets are not uniform inside. Normally, they are densest at the center and have decreasing density outward toward the surface. Model a spherically symmetric planet, with the same radius as earth, as having a density that decreases linearly with distance from the center. Let the density be 15.0E3 kg/m³ at the center and 2.0E3 kg/m³ at the surface. What is the acceleration due to gravity at the surface of this planet?
Relevant Equations
ρ = M/V ; g = GM/R²
Good evening,

I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context.
I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this:
$$ \rho(r) = \rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r $$In order to calculate the acceleration due to gravity at the surface, I know that the mass of this planet is required. My plan was to start from the density equation (with the variables being functions of radial distance from the center): $$ \rho(r) = \frac {m(r)}{V(r)} $$Then multiply both sides by the volume as a function of r: $$ \rho(r) \cdot V(r) = m(r) $$Followed by taking a derivative of both sides with respect to r: $$ \frac {d}{dr} (\rho(r) \cdot V(r)) = \frac {d}{dr}(m(r)) $$This leads to:
$$ \frac {d\rho(r)}{dr} \cdot V(r) + \rho(r) \cdot \frac {dV(r)}{dr} = \frac {dm(r)}{dr}$$Normally in the case of constant density, we would trash the first term on the LHS and move forward with the rest. However, since I have the function for the density in terms of r, I figured keeping this term would be necessary.

My next line looked like this: $$ \frac {d}{dr}(\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r) \cdot (\frac {4}{3}\pi r^3) + (\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r) \cdot \frac {d}{dr} (\frac {4}{3} \pi r^3) = \frac {dm(r)}{dr} $$ Which becomes: $$ (- \frac {\rho_{center} - \rho_{surface}} {R_{earth}}) \cdot (\frac {4}{3}\pi r^3) + (\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r) \cdot ( 4\pi r^2) = \frac {dm(r)}{dr} $$ At this point, I multiplied both sides by dr, then integrated r from 0 to R_earth and the mass from 0 to the total mass of the planet, M: $$ \int_{r=0}^{r=R_{earth}} [\frac {4}{3}\pi r^3(\frac {\rho_{surface} - \rho_{center}} {R_{earth}}) + 4\pi r^2(\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r)]dr = \int_{m=0}^{m=M} dm(r) $$After integration and a long series of algebra, I ended up with this: $$ M = \frac {4}{3} \pi \rho_{surface} R_{earth}^3 \approx 2.18 \cdot 10^{24} kg $$ Unfortunately, this result is a factor of roughly 2.6 times smaller than the correct value: $$ M_{correct} \approx 5.71 \cdot 10^{24} kg $$ If I go back to the beginning and use ##\rho(r)dV(r) = dm(r)## instead, everything works out beautifully with no trouble. My question is... why?

In problems similar to this with constant density, it was abundantly clear that ##d\rho V## was zero because the derivative of a constant density is zero. But here, the derivative of ##\rho## is non-zero. So... aren't we obligated to include it?
 
Physics news on Phys.org
TRB8985 said:
Homework Statement: Planets are not uniform inside. Normally, they are densest at the center and have decreasing density outward toward the surface. Model a spherically symmetric planet, with the same radius as earth, as having a density that decreases linearly with distance from the center. Let the density be 15.0E3 kg/m³ at the center and 2.0E3 kg/m³ at the surface. What is the acceleration due to gravity at the surface of this planet?
Relevant Equations: ρ = M/V ; g = GM/R²

Good evening,

I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context.
I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this:
$$ \rho(r) = \rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r $$In order to calculate the acceleration due to gravity at the surface, I know that the mass of this planet is required. My plan was to start from the density equation (with the variables being functions of radial distance from the center): $$ \rho(r) = \frac {m(r)}{V(r)} $$Then multiply both sides by the volume as a function of r: $$ \rho(r) \cdot V(r) = m(r) $$Followed by taking a derivative of both sides with respect to r: $$ \frac {d}{dr} (\rho(r) \cdot V(r)) = \frac {d}{dr}(m(r)) $$This leads to:
$$ \frac {d\rho(r)}{dr} \cdot V(r) + \rho(r) \cdot \frac {dV(r)}{dr} = \frac {dm(r)}{dr}$$Normally in the case of constant density, we would trash the first term on the LHS and move forward with the rest. However, since I have the function for the density in terms of r, I figured keeping this term would be necessary.

My next line looked like this: $$ \frac {d}{dr}(\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r) \cdot (\frac {4}{3}\pi r^3) + (\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r) \cdot \frac {d}{dr} (\frac {4}{3} \pi r^3) = \frac {dm(r)}{dr} $$ Which becomes: $$ (- \frac {\rho_{center} - \rho_{surface}} {R_{earth}}) \cdot (\frac {4}{3}\pi r^3) + (\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r) \cdot ( 4\pi r^2) = \frac {dm(r)}{dr} $$ At this point, I multiplied both sides by dr, then integrated r from 0 to R_earth and the mass from 0 to the total mass of the planet, M: $$ \int_{r=0}^{r=R_{earth}} [\frac {4}{3}\pi r^3(\frac {\rho_{surface} - \rho_{center}} {R_{earth}}) + 4\pi r^2(\rho_{center} - \frac {\rho_{center} - \rho_{surface}} {R_{earth}} \cdot r)]dr = \int_{m=0}^{m=M} dm(r) $$After integration and a long series of algebra, I ended up with this: $$ M = \frac {4}{3} \pi \rho_{surface} R_{earth}^3 \approx 2.18 \cdot 10^{24} kg $$ Unfortunately, this result is a factor of roughly 2.6 times smaller than the correct value: $$ M_{correct} \approx 5.71 \cdot 10^{24} kg $$ If I go back to the beginning and use ##\rho(r)dV(r) = dm(r)## instead, everything works out beautifully with no trouble. My question is... why?

In problems similar to this with constant density, it was abundantly clear that ##d\rho V## was zero because the derivative of a constant density is zero. But here, the derivative of ##\rho## is non-zero. So... aren't we obligated to include it?
The mass doesn't equal ##\rho V##, it equals the ## \int \rho ~dV ##. When density is constant ##\rho## comes outside of the integral to yield the formula ## \rho V##. So basically, you started with a false relationship for the mass, and it's still false after differentiation.
 
In $$\rho(r) = \frac {m(r)}{V(r)}$$ on LHS you have a density at radius ##r## while on the RHS you have an average density of a ball of radius ##r##.
 
Ahh, now I see what's meant by starting with a false relationship! I'd totally agree, that's not right at all.

So really, it sounds like I should be doing this instead: $$ \rho(r) = \frac {dm(r)}{dV(r)} $$ .. and not even worry about a ##d \rho \cdot V## appearing at all.
 
I suggest you look for the integral you need to evaluate: using the law of gravitation to give the surface gravity for a spherically symmetric object.

Hint: Newton's shell theorem.
 
Last edited:
I multiplied the values first without the error limit. Got 19.38. rounded it off to 2 significant figures since the given data has 2 significant figures. So = 19. For error I used the above formula. It comes out about 1.48. Now my question is. Should I write the answer as 19±1.5 (rounding 1.48 to 2 significant figures) OR should I write it as 19±1. So in short, should the error have same number of significant figures as the mean value or should it have the same number of decimal places as...
Thread 'A cylinder connected to a hanging mass'
Let's declare that for the cylinder, mass = M = 10 kg Radius = R = 4 m For the wall and the floor, Friction coeff = ##\mu## = 0.5 For the hanging mass, mass = m = 11 kg First, we divide the force according to their respective plane (x and y thing, correct me if I'm wrong) and according to which, cylinder or the hanging mass, they're working on. Force on the hanging mass $$mg - T = ma$$ Force(Cylinder) on y $$N_f + f_w - Mg = 0$$ Force(Cylinder) on x $$T + f_f - N_w = Ma$$ There's also...

Similar threads

Back
Top