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Homework Statement
The Attempt at a Solution
I already know how to do a), but what I am wondering is what the question means by expressing position in the terms of those unit vectors.
Think of polar coordinates as radial lines out from the origin (the lines of constant θ) and concentric circles surrounding the origin (the curves of constant r). The unit vector [itex]\hat{e}_{θ}[/itex] is tangent to these circles at each point.Actually I think the position would just be 5 [itex]\hat{r}[/itex] since at any point its radial position would be 5. In my dynamics class we also used the equation [itex]\vec{r}[/itex]=r [itex]\hat{e}[/itex][itex]_{r}[/itex]. [itex]\hat{e}[/itex][itex]_{θ}[/itex] was only used with velocity and acceleration equations because those would change with θ.
Yes.So the angle between the radius and the x axis is 2t?
Yes. To do this in cartesian coordinates is no problem, but, if you are going to do it in polar coordinates, you have to remember to include terms involving the derivatives of the unit vectors.So to get the velocity and acceleration vectors would I just take the derivative and second derivative of the position equation?
Yes. Exactlly.so another question is asking me to express velocity in terms of Er and Eθ, so is this what you are saying, the derivative of the unit vectors too?
Not exactly. Each or these two unit vectors changes direction with θ (i.e., are functions of θ). So, their time derivatives are their spatial derivatives with respect to θ times the derivative of θ with respect to time:When you say derivative of a unit vector do you mean the derivative of the unit vector with respect to time? If so would it be something like dEr/dt? and dEθ/dt?
You're going to kick yourself.I know how to get velocity in polar coordinates if it was cosθ and sinθ, but it is 2t so how do you take the derivative of 2t with respect to t?
Before you take the derivatives of the position vector, you need to know the derivatives of the unit vectors. It is easiest to understand, if you recall that [itex]\hat e_r =\cos(\theta) \hat i + \sin (\theta ) \hat j[/itex] and [itex]\hat e_{theta} =-\sin(\theta) \hat i + \cos (\theta) \hat j[/itex]. (You see, they are orthogonal unit vectors.)so another question is asking me to express velocity in terms of Er and Eθ, so is this what you are saying, the derivative of the unit vectors too?