Centripetal acceleration and radius

[knattagh]

I am curious how to tell how centripetal accel. changes with radius since there are two equations I can look at

a = v2/r
or
a=wr

I read on a thread that if v is constant then use the top equation and if w is constant then use the bottom equation. Is this true? If so, can you please give me a concrete example of two situation when we would use one over the other? Arent both w and v constant in uniform circular motion?!

Any advice to clear up my confusion would be much appreciated, thanks

 

[vanhees71]

Well, let's derive it. If the particle runs with constant angular velocity on a circle, we can parametrize it's trajectory as
$$\vec{x}(t)=r \begin{pmatrix} \cos(\omega t) \\ \sin(\omega t) \end{pmatrix}.$$
The velocity is the derivative wrt. time:
$$\vec{v}(t)=\dot{\vec{x}}(t)=r \omega \begin{pmatrix} -\sin(\omega t) \\ \cos(\omega) \end{pmatrix}.$$
From this you get the magnitude of the velocity (speed)
$$v=|\vec{v}|=r \omega.$$
The acceleration is given by the time derivative of the velocity,
$$\vec{a}(t)=\dot{\vec{v}}(t)=\ddot{\vec{x}}(t)=-\omega^2 r \begin{pmatrix} \cos(\omega t) \\ \sin(\omega t) \end{pmatrix}$$
and its magnitude
$$a=|\vec{a}|=r \omega^2.$$
Now you can use the speed instead of $\omega$,
$$v=\omega r \; \Rightarrow \; \omega=\frac{v}{r} \; \Rightarrow \; a=r \omega^2 = r \left (\frac{v}{r} \right )^2=\frac{v^2}{r}.$$
As you see, the magnetitudes of the velocity and acceleration are constant as long as the angular velocity $\omega$ is constant, and you can express these quantities by the others as needed.

 

[jbriggs444]

Both ω and v are constant in any particular instance of circular rotation. But then so is r. The distinction between which can be "held constant" applies when considering families of instances of circular rotation.

Since v = rω, both equations are true regardless of the situation. However, one equation or the other may be more immediately useful.

Case 1: Let's say that we have a circular race track and a time that we want to beat for 500 laps. If we want to minimize required traction, is it better to take an inside track or an outside track?

In this case, angular velocity (omega) is the constant that we are concerned with. 500 laps in the specified time. a=ωr tells us that acceleration is proportional to r. The inside track minimizes r and thereby minimizes requirements for traction.

Case 2: Let us say that we have a circular [section of an] off-ramp from a super-highway. Cars will be taking this off-ramp at highway speeds. If we want to minimize the required lateral acceleration, should the ramp curvature be tighter or broader?

In this case, linear velocity (v) is being held constant. Highway speeds. a=v²/r tells us that acceleration is inversely proportional to r. A broader curve maximizes r and thereby minimizes the lateral acceleration.

 

[knattagh]

Thanks you for your replies!

jbriggs, can you please elaborate on why w and v are held constant in cases 1 and 2 respectively.

Also, how do the two cases differ physically? It seems to me like the same thing is happening in both cases: a car is moving in a circle.

Thanks!

 

[jbriggs444]

jbriggs, can you please elaborate on why w and v are held constant in cases 1 and 2 respectively.

Also, how do the two cases differ physically? It seems to me like the same thing is happening in both cases: a car is moving in a circle.

The two cases do not differ physically. They are the same. It is the question we are trying to answer that is different:

For a given angular velocity, how does acceleration vary with radius?
For a given velocity, how does acceleration vary with radius?

 

[knattagh]

but both times isn't it given that both v and w are constant ? I know you only explicitly stated that either one or the other is "held constant" but isn't it true that both v and w will be constant in each situation? (when I say v I mean speed by the way).

 

[Philip Wood]

The two excellent cases that jbriggs444 has provided each encompasses an infinity of different "situations"…

In the first case, we're considering cars with a fixed $\omega$ because they're doing 500 laps in a fixed time. The different "situations" are cars doing their laps at different radii. Cars doing laps with bigger r will have to have bigger v in order to have the same $\omega$ as cars doing laps with smaller r.

In the second case, we're considering cars with a fixed v because they're going at 'highway speed'. The different "situations" are cars doing turns of different radii. Cars doing tighter turns (smaller r) will have to have larger $\omega$.

There are lots of cases in Physics where a question seems to have two (or more) contradictory answers unless you specify carefully what's being kept constant. Here's another well known example: a 10 $\Omega$ and a 20 $\Omega$ resistor are put, one at a time, in the same gap in a circuit. Which will dissipate more power? [Hint... Let one circuit be simply a 12 V battery (of negligible internal resistance) with connecting wires coming from each of its terminals. Let the other circuit be a 12 V battery with a 100 k$\Omega$ resistor in series with it, and two connecting wires.]

 

[BruceW]

the definition of uniform circular motion is that the angular (and linear) speed are constant. If the object was just in "circular motion", then the angular (and linear) speed are not necessarily constant.

 

[knattagh]

I think I understand now. If we hold one constant (either v or w) while we change r, the other (w or v) has to change. Therefore we can only use one of the equations while r is changing.

Thanks you all for your helpful comments!