a=f(dn/dt)

[Cristiano]

I know the value of dn/dt (e.g. dn/dt = 123).

a³ = 345 / n²

Is there any way to calculate da/dt?

 

[taeto]

Yes. And somewhat unremarkably \( dy/dt = dy/dx \times dx/dt \) holds, under reasonable assumptions, so that in this case

\[ da/dt = d( \sqrt[3]{345/n^2} )/dt = d( \sqrt[3]{345} \times n^{-2/3} )/dt = \]

\[ \sqrt[3]{345}\times d( n^{-2/3} )/dn\times dn/dt = \sqrt[3]{345}\times -2n^{-5/3}/3\times  123 = -82\sqrt[3]{345}/\sqrt[3]{n^5}. \]

Edited August 4, 2018 by taeto

[Cristiano]

If I put the real numbers, I get a wrong result. Could you please confirm that the formula is correct?

[taeto]

10 minutes ago, Cristiano said:

If I put the real numbers, I get a wrong result. Could you please confirm that the formula is correct?

Can you present those real numbers?

[Cristiano]

The original formula is a³ = 398600.8 / (2 * pi * n)²

With dn/dt= 6.07340239 * 10^-12 rad/s² I get da/dt= -1.51970883 * 10^-5 km/s² which is -113445.6 km/day², while I expect about -1 to -10 km/day².

Edited August 4, 2018 by Cristiano

[taeto]

46 minutes ago, Cristiano said:

The original formula is a³ = 398600.8 / (2 * pi * n)²

With dn/dt= 6.07340239 * 10^-12 rad/s² I get da/dt= -1.51970883 * 10^-5 km/s² which is -113445.6 km/day², while I expect about -1 to -10 km/day².

So you got da/dt is not da/dn x dn/dt? I cannot quickly see through the conversion between seconds and days here, sorry.

[Cristiano]

I know a, n and dn/dt. a³= 10096.676 / n². I need to calculate da/dt.

398600.8 has unit km³/s², n is rad/s, dn/dt is rad/s².

EDIT: Probably I'm wrong in the unit of measurement of da/dt; I thought it was km/s², but it should be km/s, right?

Edited August 4, 2018 by Cristiano

[taeto]

On 8/5/2018 at 12:56 AM, Cristiano said:

I know a, n and dn/dt. a³= 10096.676 / n². I need to calculate da/dt.

398600.8 has unit km³/s², n is rad/s, dn/dt is rad/s².

EDIT: Probably I'm wrong in the unit of measurement of da/dt; I thought it was km/s², but it should be km/s, right?

The dimensions do not fit together. From the first equation follows that a is a length. Hence da/dt is a speed and should be measured in m/s or km/s as you also added.

Note that angles are dimensionless, so although it is nice of you to point out that you measure in radians, the true unit for angular speed n is 1/s, and the unit for angular acceleration is 1/s².

 With your numbers, in units of km and s, I get \( da/dt \approx -8.75\cdot 10^{-11} n^{-5/3} \) with unit km/s.

In your equation for \(a^3\) you have \( (2\pi n)^2 \) in the denominator, are you sure that it should not be just \( n^2, \) since \(n\) is already measured in rad/s?

 

Edited August 6, 2018 by taeto

[Cristiano]

You're right (but I need 2*pi because my original n is in revolution/day).

What about da/dt= -2 * 10096.676 / n³ * dn/dt / (3 * a²)

?

[taeto]

On 8/4/2018 at 11:47 PM, Cristiano said:

The original formula is a³ = 398600.8 / (2 * pi * n)²

1 hour ago, Cristiano said:

You're right (but I need 2*pi because my original n is in revolution/day).

What about da/dt= -2 * 10096.676 / n³ * dn/dt / (3 * a²)

 

On 8/5/2018 at 12:56 AM, Cristiano said:

I know a, n and dn/dt. a³= 10096.676 / n². I need to calculate da/dt.

398600.8 has unit km³/s², n is rad/s, dn/dt is rad/s².

Fine, we can try to put the things together that are in various units. Quite pleased to see that we avoid to consider nautical miles, imperial pounds, fluid barrels etc.

The original formula reads, when units are included:

\[ a^3 = 398600.8 \mbox{ km}^3/\mbox{s}^2 / (2\pi n \mbox{ day}^{-1})^2. \]

Using \( 1 \mbox{ km } = 10^3 \mbox{ m }\) and \( 1 \mbox{ day }=  8.64*10^4 \mbox{ s}: \)

\[ a^3 = 398600.8 * 10^9 \mbox{ m}^3/\mbox{s}^2 / (2\pi n * ( 8.64*10^4 \mbox{ s})^{-1})^2 \approx  7.53713*10^{22}/n^2 \mbox{ m}^3.\]

I assume that since n isn't actually measured in rad/s, also the value that you gave for dn/dt isn't really measured in rad/s^2. Whatever unit it is, if you can figure out da/dn in units of m from this equation, then multiply by dn/dt in units of 1/s (revolutions per second), then the product ought to give the answer in m/s. 

 

 

Edited August 6, 2018 by taeto