How Earth's rotation is changed by the change in inertia when you get out of bed

[robax25]

Homework Statement

Please consider what might happen if you got up early in the morning! Do not focus on your most respected tasks during the day, but just calculate how the earth rotation is changed as you are changing the inertia by getting up.

Relevant Equations

Earth moment of inertia I = 4/3 pi r³ r= Radius of the earth. =6.3 * 10^6 m.

angular velocity w = V/r

Don't understand from where should I start?

 

[berkeman]

Hmm, that is a pretty goofy question all right. Well, I'd start with the equation for your MOI with respect to the center of the Earth, and calculate the numbers for laying down and standing up (how does your center of mass change?). Then calculate the MOI of the Earth assuming it is spherical (it is not) and of uniform density (it is not). Then calculate the change in angular velocity of the Earth when you add in your change of MOI. Hopefully your calculator has a lot of digits in the display...

 

[jbriggs444]

Hmm, that is a pretty goofy question all right. Well, I'd start with the equation for your MOI with respect to the center of the Earth, and calculate the numbers for laying down and standing up (how does your center of mass change?). Then calculate the MOI of the Earth assuming it is spherical (it is not) and of uniform density (it is not). Then calculate the change in angular velocity of the Earth when you add in your change of MOI. Hopefully your calculator has a lot of digits in the display...

If you do the calculations in the right order and if your calculator does scientific notation then the number of digits need not be a problem.

One key trick is not to subtract two large and nearly equal numbers from one another.

So, for instance, instead of calculating your moment of inertia about the center of the Earth before getting up and your moment of inertia about the center of the Earth after getting up and subtracting the one from the other, calculate the change in your moment of inertia due to getting up.

Similarly, do not try to find the moment of inertia of the Earth before and after and compute the rotation rates of the Earth before and after. Instead, figure out the percentage change in the moment of inertia of the Earth and the resulting percentage change in its rotation rate.

Done well, you would not even need a calculator. The back of an envelope would do for a little algebra and a little computation.

Doing this sort of thing to reduce the impact of truncation and round-off errors in calculations is part of what "numerical analysis" is all about.

 

[robax25]

Hmm, that is a pretty goofy question all right. Well, I'd start with the equation for your MOI with respect to the center of the Earth, and calculate the numbers for laying down and standing up (how does your center of mass change?). Then calculate the MOI of the Earth assuming it is spherical (it is not) and of uniform density (it is not). Then calculate the change in angular velocity of the Earth when you add in your change of MOI. Hopefully your calculator has a lot of digits in the display...

As I need to calculate the mass moment of inertia with respect to Earth's radius .It means I need to use the formula. I = I + Mr² Here, M=80kg
= 5.04 * 10^8 kg-m².

r(earth)= 6.3* 10^6 m
M=80 kg
I =0.5 Mr²
r=40cm =0.4m

My question is that is it ok?
Second question is that where I use center of mass If I am laying and If I am standing up.I will calculate it but why?

 

[hutchphd]

First your equation for the moment of inertia of the Earth is woefully incorrect (the one for your mass is better).
Here is my stupid answer: my bed is the same height above the floor as my center of mass while standing. So getting out of bed will not change the rotational speed of the earth. Of course that is not the "expected" answer but it is correct. Of course if you go down stairs (as we might do here) then you are in difficulty.

 

[berkeman]

my bed is the same height above the floor as my center of mass while standing.

 

[Keith_McClary]

It should be noted that moving north or south at the same altitude changes your MoI, so it depends which side of the bed you got out of.

 

[robax25]

Hmm, that is a pretty goofy question all right. Well, I'd start with the equation for your MOI with respect to the center of the Earth, and calculate the numbers for laying down and standing up (how does your center of mass change?). Then calculate the MOI of the Earth assuming it is spherical (it is not) and of uniform density (it is not). Then calculate the change in angular velocity of the Earth when you add in your change of MOI. Hopefully your calculator has a lot of digits in the display...

How do I calculate my MoI with respect to the earth? As I already have calculated, I got replied that it is wrong. How do I do that?

 

[Keith_McClary]

How do I calculate my MoI with respect to the earth?

First let's assume you are at the equator. Then your MoI is $mr^2$ and when you stand up it is $mR^2$, where $R$ is slightly larger.

 

[robax25]

mr²= 80kg * (6.3*10^6 m)² =3.1752*10^15 kg-m².
MR²=80kg*(6300001.7 m)²=3.17501714*10^15 k-m² and I just add 1.7m

They are almost same.

 

[Keith_McClary]

OK, I don't like typing out long numbers with units, so don't plug in the numbers till the end. What is the equation for conservation of angular momentum if the angular velocities are $\omega$ and $\Omega$ ?

 

[robax25]

L=I*omega and Eath angular velocity Ω = 7.272 *10^-5 rad /s Here Ω=2*pi/24 h

I can write
Iω=IΩω= 7.2716*10^-15 rad/s at the end, it is a very small value.

 

[Keith_McClary]

So, equate the before and after expressions for $L$.

 

[robax25]

I did it

 

[PeroK]

If this was my homework, I think it would be a day to go back to bed!

 

[jbriggs444]

mr²= 80kg * (6.3*10^6 m)² =3.1752*10^15 kg-m².
MR²=80kg*(6300001.7 m)²=3.17501714*10^15 k-m² and I just add 1.7m

They are almost same.

Let us try to follow the suggestion by @Keith_McClary. Do not plug in numbers until later. Do it with algebra.

The assumption you are making here is that upon getting out of bed, your center of mass rises by 1.7 meters. That assumption is almost certainly wrong, but let's go with it. Let us call this increase in altitude $h$.

So now we are trying to compare $mR^2$ with $m(R+h)^2$.

If you think back to math class, that latter formula can be expanded as $mR^2 + 2mRh + mh^2$.

If one subtracts $mR^2$ from $mR^2 + 2mRh + mh^2$ the result is $2mRh + mh^2$. That is how much moment of inertia you've added by getting out of bed.

That result can be computed without subtracting large and precise numbers from one another. One might also note that the $mh^2$ term is negligible by comparison to the $2mRh$ term. It can be safely ignored.

 

[Keith_McClary]

If one subtracts

But we want the ratio. And then, do we need to modify the formula for nonzero latitudes (using the distance from the Earth's axis, not the center)?

 

[jbriggs444]

But we want the ratio. And then, do we need to modify the formula for nonzero latitudes (using the distance from the Earth's axis, not the center)?

Yes, we do need the latitude. We need to be able to convert from the height increase to an increased distance from the axis. [And when we get to the algebra, we need to distinguish $R$ as the radius of the Earth from $r$ as the our bed's distance from the axis of rotation. Which may mean paying even more attention the the algebra I walked through above].

We do not need the ratio. What we are after is the change in rotational period^(*). We do not need a number for the ratio in order to compute that. We can continue to follow your advice and use algebra rather than calculation.

Edit: The ratio we do not need is Earth moment of inertia before to Earth moment of inertia after. That is a number extremely close to one. A number with a lot of significant digits. Hard to calculate with. Possibly you mean the ratio of the increase in moment of inertia from you getting out of bed to the total moment of inertia of the earth. That is a ratio that is very close to zero. A small number without a lot of significant digits. Easy to calculate with. My apologies if I misinterpreted you above.

(*) As I interpret the original question, we are interested in how much longer the rotational period of the Earth is after we rise from bed compared to how long it was prior. One would expect an answer as some small fraction of a second or, perhaps, as some very small fraction of a day.

 

[vela]

Hmm, that is a pretty goofy question all right. Well, I'd start with the equation for your MOI with respect to the center of the Earth, and calculate the numbers for laying down and standing up (how does your center of mass change?).

Second question is that where I use center of mass If I am laying and If I am standing up.I will calculate it but why?

The word you want here is lying unless you meant to imply you were, say, laying an egg.