What Factors Influence Acceleration in Non-Uniform Circular Motion?

[MasterTinker]

I know this problem looks easy but my physics mind is out of shape.

Problem

Imagine that you have a car traveling along a circular, horizontal track or radius $r$, with tangential acceleration $a_{lin}$. If the car begins moving around the track with velocity 0 m/s:

1. What is the tangential speed of the car after one lap?

2. What is the acceleration of the car after it completes one lap?

3. What is the $$\displaystyle\lim_{r\rightarrow\infty}$$ of the answer in 2?

Attempt at a Solution

1.

In order to calculate the tangential speed of the car after one lap, $v_{lap}$, I first calculate the time it would take for it to complete a lap. The distance around the track is $2\pi r$, and using a kinematics equation:

$$2\pi r=\frac{1}{2}a_{lin}t^2$$

$$t=2\sqrt{\frac{\pi r}{a_{lin}}}$$

$v_{lap}$ is just acceleration by time, therefore:

$$v_{lap}=a_{lin}t$$

$$v_{lap}=a_{lin}\left(2\sqrt{\frac{\pi r}{a_{lin}}}\right)$$

$$v_{lap}=2\sqrt{\pi a_{lin}r}$$

2.

If I define $\hat{r}$ to be the unit vector pointing directly outwards from the center of the circle, and $\hat{v}$ to be the unit vector pointing in the direction of the car's tangential motion, then the acceleration of the car, $a_{car}$, is:

$$a_{car}=-a_{centripetal}\hat{r}+a_{tangential}\hat{v}$$

The centripetal acceleration is proportional to the car's velocity and the radius of the track, while the tangential acceration is just $a_{lin}$, therefore:

$$a_{car}=-\frac{v_{lap}^2}{r}\hat{r}+a_{lin}\hat{v}$$

$$a_{car}=-\frac{\left(2\sqrt{\pi a_{lin}r}\right)^2}{r}\hat{r}+a_{lin}\hat{v}$$

$$a_{car}=-4\pi a_{lin}\hat{r}+a_{lin}\hat{v}$$

3.

Uh-oh

What's Wrong?

I don't see any problem with the way I derived the velocity of the car after one lap. I think how I did the car's acceleration is okay, but what I don't understand is that it is not based on the radius of the track (does this seem weird to anyone else?) and I can't figure out how to interpret the limit required in part 3.

What I would think is that as $r$ approaches $\infty$ the car will have an infinite speed by the time it completes a lap, and therefore have an infinite centripetal acceleration. Or would the car not be able to complete a lap? Or would the centripetal acceleration be 0 as a circle of infinite radius becomes a line? I really don't know what to think, please help me interpret this problem.

 

[MasterTinker]

By the way I apologize for not placing this in the homework forum. If it's crowding then please redirect it there : )

 

[baba89]

Non-uniform circular motion can be a tricky concept to grasp, especially if you're out of practice. Your attempt at a solution is on the right track, but there are a few issues that need to be addressed.

Firstly, your calculation for the tangential speed after one lap is correct, but your interpretation of the limit in part 3 is incorrect. As the radius of the track approaches infinity, the tangential speed of the car will also approach infinity, but this does not mean that the car will have an infinite speed. The car will still have a finite speed, but it will be continuously increasing as it completes each lap. This is because the tangential acceleration, a_lin, is constant and not affected by the radius of the track.

Secondly, your calculation for the acceleration of the car after one lap is incorrect. The centripetal acceleration, a_{centripetal}, is not proportional to the car's velocity, but rather to the square of its velocity. This means that the correct equation for the car's acceleration is:

a_{car}=-\frac{v_{lap}^2}{r}\hat{r}+a_{lin}\hat{v}

This will give you a different answer for the acceleration after one lap, and a different interpretation for the limit in part 3.

In terms of interpreting the limit, as the radius of the track approaches infinity, the car will essentially be traveling in a straight line. This means that the centripetal acceleration will approach zero, as there will be no curvature in the motion. However, the tangential acceleration, a_lin, will still be present and will continue to increase the car's speed. Therefore, the limit in part 3 is 0, as the centripetal acceleration approaches 0, but the tangential acceleration remains constant.

I hope this helps clarify your understanding of non-uniform circular motion and how to approach this problem. It's important to always double check your equations and interpretations to ensure the correct solution. Keep practicing and your physics mind will be back in shape in no time!