What does "in a ... reference frame" mean?

Leonard: Actually, it's centripetal force, which is an inward force generated by the glass acting on the olive. (The olive drops.) Excuse me. (Leonard disappears under table.)Now, if you were riding on the olive, you'd be in a non-inertial reference frame, and would (he bangs his head on the underside of the table.)

This is an explanation of basic Newtonian physics. When Leonard talks about reference frame, he is talking about… hmm... Your reference frame is your point of view. If you are in a non-inertial reference frame, then you are in a place where from your point of view there is no inertia. I think what he's referring to is that if you were to fall off the olive, you would travel along a path tangential to the circular path you were traveling, rather than continue your motion as it was. This is all best explained with a picture.

"A non-inertial reference frame is a frame of reference that is undergoing acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will in general detect a non-zero acceleration."

Ohhh! Okay. I get it. So... how can I explain this

If you are riding the olive, that means you are at rest on it, even though the olive is not at rest with respect to the inertial frame, or the frame where everything is still, earth. But despite you being at rest on this "plane" in this case the "olive" you will still feel acceleration (dv/dt). Where as when you are at rest in the inertial plane, e.g. earth, you don't feel acceleration.

In any case, the point here is, Leonard was going into complex discussions and he bumped his head, bringing him back to the less complex.

I agree. It's tough. Un...物理です。I want to draw pictures to explain it xD

あはははは! 勿論！
(いえいえ。高校生です。笑)

"Reference frame" is physics terminology, from the theory of relativity. In short, it's similar to "point of view" or "basis for comparison". A longer explanation:

Einstein's discovery was that classical Newtonian physics works perfectly only when there is no acceleration in a system. As acceleration increases, from the point of view of an accelerating observer, mass seems to increase, length seems to shrink, and time seems to slow down.

Now consider you and me as two observers, let's say on two planets in space. If the planets are getting farther apart, does that mean I am accelerating away from you, or you are accelerating away from me? The answer to this matters, since the one who is accelerating will observe these changes in mass, length, and time.

In order to answer that, we would need some third object to use for comparison. We might call this a "reference" object, since we refer to it in order to make our comparisons. We should find something that is absolutely stationary to use as a reference, since then everyone can make calculations relative to that object, and we can all agree on who is accelerating and who isn't.

But the key idea of relativity is that there turns out to be no such stationary reference. There is not only no object that we can call "stationary", there isn't even a theoretical point of view that we can use as a basis for comparison. Instead, every observer's point of view (the observer's "reference frame") is equally valid as a basis for observations and calculations. In relativity this is stated as "there is no privileged reference frame".

Now, this means that two observers in different reference frames will arrive at different answers if they try to calculate the properties of things they observe. This is different from Newtonian physics, according to which everyone observing something should arrive at the same answers about the forces and mass and inertia involved. But if nothing in the system is accelerating, Newtonian calculations still work just fine. Such a system all shares the same "reference frame", since nothing in it is accelerating. Such a reference frame is often called an "inertial reference frame", since "inertia" was a key part of Newton's laws.

So any reference frame undergoing acceleration is a non-inertial reference frame, like the olive. There is a force acting on it from the glass, so it undergoes centripetal acceleration. The joke is that the relativistic effects of acceleration are very very tiny, so tiny they won't matter unless acceleration is huge. It really doesn't matter that the olive is in a non-inertial reference frame, but Leonard is very precise. Yet at the same time he is being so extremely careful about physics, he is so clumsy that he hits his head.

Jmstewy is right. By the way, you should disregard my first answer. This has little to do with circular paths.

本当に大丈夫よ。

いや。日本語下手ですよ

良かったですね！

いいえ、アニメじゃない。高校でクラスがあったり、自分で勉強したりする。アニメを見たから、少し習ったけど、今学校が忙し過ぎる。アニメの暇はないねf^_^;)

^文法がすみません