Deriving the Differential Position Vector in Cylindrical Coordinates

[Istiak]

Homework Statement

differential position vector

Relevant Equations

vectors, differentiation

I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an equation which came from logic and mind rather than deriving?

 

[PeroK]

Homework Statement:: differential position vector
Relevant Equations:: vectors, differentiation

I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an equation which came from logic and mind rather than deriving?

This is very similar to the question you asked previously about cylindrical coordinates. You should be able to derive that expression either geometrically (by drawing a diagram) or algebraically (by converting from Cartesian).

 

[Istiak]

This is very similar to the question you asked previously about cylindrical coordinates. You should be able to derive that expression either geometrically (by drawing a diagram) or algebraically (by converting from Cartesian).

What I could draw that is It :

I think it wasn't correct graph. (cylindrical was little bit harder that's why I am not taking a look at that.)

 

[Istiak]

Ow! I got a figure in book.

 

[PeroK]

Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$

This is the differential increment in the position vector: How a small change in $\vec r$ relates to small changes in $r, \theta, z$.

Note that it should be $d\vec r$ on the left hand side.

 

[Istiak]

This is the differential increment in the position vector: How a small change in $\vec r$ relates to small changes in $r, \theta, z$.

Note that it should be $d\vec r$ on the left hand side.

got the answer... But, book wrote that; that's dr on the LHS.

 

[PeroK]

got the answer... But, book wrote that; that's dr on the LHS.

The RHS is a vector, so the LHS must be too.

 

[Delta2]

Yes the differential position vector comes from the derivative (with respect to time) of the position vector.
The position vector in cylindrical coordinates is $$\vec{r}=r(t)\hat r(t)+z(t)\hat z$$.

Take the derivative of that with respect to time and have in mind that $\hat r(t)=\cos\theta(t)\hat x+\sin\theta(t)\hat y$. Also have in mind that $r'(t)dt=dr,\theta'(t)dt=d\theta,z'(t)dt=dz$ and that the unit vector $\hat \theta(t)=-\sin\theta(t)\hat x+\cos\theta(t)\hat y$.

 

[Delta2]

Ah btw, I found a mini problem both with the OP notation and mine notation at post #8, we use the same letter $r$ to denote the position vector and the radius in cylindrical coordinates, i should 've written $$\vec{r}=\rho(t)\hat \rho(t)+z(t)\hat z$$, and the OP I think should write $$d\vec{r}=d\rho\hat\rho+\rho d\theta\hat\theta+dz\hat z$$