Radial and Circumferential Components in terms of t

[Trapezoid]

Homework Statement

Let $\vec{R} = (t + \sin t) \hat{i} + (t + \cos t) \hat{j}$ denotes at time t the position of a moving particle. Determine the radial and circumferential components of acceleration as a function of t.

Homework Equations

$$v_r = \dot{r}$$
$$v_{\theta} = r\dot{\theta}$$

The Attempt at a Solution

I tried to write r in terms of t using $r = \sqrt{x^2 + y^2}$, but the derivative was complicated. I do not know how to write $\theta$ in terms of t. We were asked to determine the tangential and normal components of acceleration in the previous part of the question, but I do not see how they will avail me. Could anybody point me in the right direction?

Thanks,
Trapezoid

 

[tiny-tim]

Hi Trapezoid!

Determine the radial and circumferential components of acceleration as a function of t.

The tangential component (i assume that's what they mean by circumferential) is just d|v|/dt

(and you know the magnitude and direction of the total acceleration)

 

[Trapezoid]

Hi tiny-tim,

It is my understanding that the tangential and circumferential components are different. When I say the circumferential component of acceleration, I refer to the acceleration in the direction of $\theta$, ie: the change in the rate of change of $\theta$. I'm having trouble finding $\theta$ as a function of t..

Does that make sense? Have I misunderstood?

Thanks,
Trapezoid

 

[tiny-tim]

When I say the circumferential component of acceleration, I refer to the acceleration in the direction of $\theta$, ie: the change in the rate of change of $\theta$. I'm having trouble finding $\theta$ as a function of t..

ah, not a terminology I've come across before

ok, then r is the position, r'' is the acceleration, r''.r/|r| is the radial component, and what's left is the circumferential component

 

[Trapezoid]

Thanks tiny-tim,

Let me make sure that I understand correctly. Is $\frac{r}{|r|}$ the unit vector for motion in the radial direction?

 

[tiny-tim]

yes, the unit vector in the radial direction is r/|r|