Obostrzenie
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Good morning.
I would like to ask you for help with understanding rearranging of equation that I've encountered while reading 'mechanics' by Kittle, Knight and Ruderman. I'm talking about this:
By reference to Eqs. (2.25) and (2.26) for \(\displaystyle d\hat{\mathbf{r}}/dt\) and \(\displaystyle d\hat{\mathbf{\theta}}/dt,\) we bring this expression into the terms:
. . .\(\displaystyle a\, =\, \dfrac{d\mathbf{v}}{dt}\, =\, \dfrac{d^2r}{dt^2}\hat{\mathbf{r}}\, +\, \dfrac{dr}{dt}\dfrac{d \theta}{dt}\hat{\mathbf{\theta}}\,+\, \dfrac{dr}{dt}\dfrac{d \theta}{dt}\hat{\mathbf{\theta}}\, +\, r\dfrac{d^2 \theta}{dt^2}\hat{\mathbf{\theta}}\, -\, r\left(\dfrac{d \theta}{dt}\right)^2\hat{\mathbf{r}}\)
Then, by collecting terms and a little rearranging, we write this in the usual fashion:
. . .\(\displaystyle a\, =\, \Bigg[\dfrac{d^2r}{dt^2}\, -\, r\left(\dfrac{d \theta}{dt}\right)^2\Bigg]\, \hat{\mathbf{r}}\, + \, \dfrac{1}{r}\, \Bigg[\dfrac{d}{dt}\, \left(r^2\, \dfrac{d \theta}{dt}\right)\Bigg] \hat{\mathbf{\theta}}\qquad (2.30)\)
Particularly I have problem with understanding the second term. I think I see how we can multiply it to get each of terms with unit vector theta from the upper equation.
That would be one way:
\(\displaystyle 1)\, \dfrac{1}{r}\Bigg[\dfrac{d}{dt}\left((r^2)\left(\dfrac{d \theta}{dt}\right)\right)\Bigg]\, =\, \dfrac{1}{r}\Bigg[(r^2)\left(\dfrac{d^2 \theta}{dt^2}\right)\Bigg]\, =\, r\, \left(\dfrac{d^2 \theta}{dt^2}\right)\)
And the other:
\(\displaystyle 2)\, \dfrac{1}{r} \Bigg[\dfrac{d}{dt}\left((r^2)\left(\dfrac{d \theta}{dt}\right)\right)\Bigg]\, =\, \dfrac{1}{r}\Bigg[\left(\dfrac{dr^2}{dt}\right) \left(\dfrac{d \theta}{dt}\right)\Bigg]\, =\, \left(\dfrac{dr}{dt}\right)\left(\dfrac{d \theta}{dt}\right)\)
But my question is how come that from 3 terms with θ unit vector from upper equation we have only one? To be honest I doubt that my multiplications are allowable but I stuck on that and don't have another idea. What am I missing?
I would like to ask you for help with understanding rearranging of equation that I've encountered while reading 'mechanics' by Kittle, Knight and Ruderman. I'm talking about this:
By reference to Eqs. (2.25) and (2.26) for \(\displaystyle d\hat{\mathbf{r}}/dt\) and \(\displaystyle d\hat{\mathbf{\theta}}/dt,\) we bring this expression into the terms:
. . .\(\displaystyle a\, =\, \dfrac{d\mathbf{v}}{dt}\, =\, \dfrac{d^2r}{dt^2}\hat{\mathbf{r}}\, +\, \dfrac{dr}{dt}\dfrac{d \theta}{dt}\hat{\mathbf{\theta}}\,+\, \dfrac{dr}{dt}\dfrac{d \theta}{dt}\hat{\mathbf{\theta}}\, +\, r\dfrac{d^2 \theta}{dt^2}\hat{\mathbf{\theta}}\, -\, r\left(\dfrac{d \theta}{dt}\right)^2\hat{\mathbf{r}}\)
Then, by collecting terms and a little rearranging, we write this in the usual fashion:
. . .\(\displaystyle a\, =\, \Bigg[\dfrac{d^2r}{dt^2}\, -\, r\left(\dfrac{d \theta}{dt}\right)^2\Bigg]\, \hat{\mathbf{r}}\, + \, \dfrac{1}{r}\, \Bigg[\dfrac{d}{dt}\, \left(r^2\, \dfrac{d \theta}{dt}\right)\Bigg] \hat{\mathbf{\theta}}\qquad (2.30)\)
Particularly I have problem with understanding the second term. I think I see how we can multiply it to get each of terms with unit vector theta from the upper equation.
That would be one way:
\(\displaystyle 1)\, \dfrac{1}{r}\Bigg[\dfrac{d}{dt}\left((r^2)\left(\dfrac{d \theta}{dt}\right)\right)\Bigg]\, =\, \dfrac{1}{r}\Bigg[(r^2)\left(\dfrac{d^2 \theta}{dt^2}\right)\Bigg]\, =\, r\, \left(\dfrac{d^2 \theta}{dt^2}\right)\)
And the other:
\(\displaystyle 2)\, \dfrac{1}{r} \Bigg[\dfrac{d}{dt}\left((r^2)\left(\dfrac{d \theta}{dt}\right)\right)\Bigg]\, =\, \dfrac{1}{r}\Bigg[\left(\dfrac{dr^2}{dt}\right) \left(\dfrac{d \theta}{dt}\right)\Bigg]\, =\, \left(\dfrac{dr}{dt}\right)\left(\dfrac{d \theta}{dt}\right)\)
But my question is how come that from 3 terms with θ unit vector from upper equation we have only one? To be honest I doubt that my multiplications are allowable but I stuck on that and don't have another idea. What am I missing?
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