Period of a Pendulum in an Accelerating Elevator | Explained & Calculated
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- Опубликовано: 29 сен 2024
- Calculate the period of a pendulum in an accelerating elevator. Rather than simply attributing the change in perceived gravity to being in an accelerated reference frame, this solution backs up a step to explain what a person inside the elevator would actually feel. Then looks at the change in the tension holding up the pendulum bob and its affect on the restoring force of the pendulum back toward equilibrium.
This is a very common problem in high school physics. It also shows up in introductory physics courses, AP Physics C Mechanics and on the JEE.
His greatness Walter Lewin also posed this problem in a series of online physics questions he posed to the RUclips community. Problem #89
man these are some really impressive editing techniques, the timing and smoothness to it
Much appreciated!
A good video with an incorrect (or, at the very least, highly misleading) explanation of the tension. The y-component of the tension and gravity do NOT cancel out - the pendulum swings along a circular arc, not a horizontal line, so the radial force components are what cancel out (in the non-inertial frame of the elevator). While your statement about vertical components cancelling out becomes true in the small-angle limit since tangential and horizontal directions become identified, it is not strictly true and can be misleading since it would not produce accurate generalizations to large angles.
Even for the standard simple pendulum, the tension F_T is given by m*(g + a_o_y)/cos(theta) where I've defined a_o_y = L*(theta_max)^2*(omega)^2*cos(theta)*sin^2(omega * t + phi) - g*sin^2(theta) (for theta in (-pi/2, pi/2), and where phi is the phase constant, theta_max the angular amplitude, omega the angular frequency). Notice that this behaves properly under various special cases (e.g., theta_max = theta = 0 gives a_o_y = 0 and F_T = m*g).
You can derive this result for F_T by first getting the angular acceleration alpha from the gravitational torque m*g*L*sin(theta) = m*L^2*alpha, then setting the tangential acceleration a_tan = L*(alpha), and then getting the y-component of that vector a_tan_y = -g*sin^2(theta), then adding that to the y-component of the radial acceleration a_rad_y = a_rad * cos(theta) with a_rad = v^2 / L where |v| = L*(theta_max)*(omega)*sin(omega * t + phi). Of course, this could be written differently if we decided to plug in omega = sqrt(g/L), btw.
In the case of the pendulum being in an elevator, we must add a term m*a to the net force in the y-direction on the pendulum bob, resulting in an increase in F_T as you said; doing this, we'll algebraically see that the solution becomes the same as if we had just replaced "g" with "(g+a)", as it must in order to match what would be predicted by an observer in the non-inertial frame of the elevator. However this increase in F_T does not happen in the way you said, nor for the reason you said.
You are completely correct however you've missed the point. 99% of the people who ever see this problem are going to be treating pendulums as simple harmonic oscillators and operating within the SAA; and the people who aren't certainly don't need me explaining simple stuff like inertial reference frames to them.
@@INTEGRALPHYSICS That's understandable up to a point, however I think it's important to lay out the caveats to such an explanation up front. Besides avoiding confusion for kids who may wonder at how something that clearly moves on a circular arc is only supposed to move horizontally (and it's an important geometric limit-case in other situations as well, such as relating circular motion to linear motion in the limit of R approaching infinity), it helps to emphasize that models are only useful within limited domains of applicability. That kind of epistemic and meta-cognitive awareness is crucial for budding young scientists/engineers to develop.
To cut it out entirely in the interest of limiting ourselves to simplified pseudo-explanations can hamper their higher-level reasoning and modeling abilities by feeding a false sense of certainty, rather than recognizing and accurately assessing the various shades of grey which are inherent in our attempts to model real phenomena. Sure, if they grasp the incomplete explanation they can answer this question on a test with some (probably over-inflated) level of belief that they truly understand it, but they are entirely blind to the frailty and misleading incompleteness of their answer. Cutting out discussions of the shades of grey inherent in considerations of each model's domain of applicability sterilizes and trivializes the subject in general, removing its vital essence and reducing it to a merely didactic question-answering game. It can be no wonder then that students, and even some academics and professionals in programs that physics serves, then start to question its worth in their curriculum/training. In short, it's a brittle trade-school education mentality as opposed to a truly empowering liberal education mentality.
Also, SHM in general presents a golden opportunity to introduce to them a sense that things they (should) learn in intro calculus, like Taylor series, are crucial to the subject. Even kids who are taking only college-level physics can gain something by learning about Taylor series and the utility of calculus via various other mentions and highlights during a course - maybe it can even help motivate them to rethink the worth of the subject. Although they themselves may not pursue it, getting at least some sense of why and how it is useful can transform their attitudes toward the subject, which can have a knock-on effect with their peers that they talk to, or with their own children in the future. They perhaps don't need the full explanation, but they deserve to know that a fuller explanation exists, as well as what tools it entails if they are interested in learning more.
So at the very least, for a video geared to a general audience with myriad reasons for being here, I would suggest that there be two levels of explanation offered; one for people just looking for a relatively superficial and narrow model, and another for those interested in looking at when, why, and how that model breaks down (and how to correct it). The latter would include prospective physics, maths, engineering, and chemistry students among others, and I think overall would constitute far more than a mere 1% of students who encounter the question. It would also give you a chance to make more content that views from this would be likely to spill-over to, so it'd be a win-win.
My two cents at least. I appreciate the work you put in here, along with other RUclipsrs, which is why I bother to make such a long (and hopefully constructive) criticism. Cheers.
Again you make a very good point and for that I thank you. I agree with you about the ramifications of distilling an explanation to its most common denominator. I will certainly keep this feedback in mind moving forward. Thanks!
It's still so confusing for me why time period decrease when elevator move upward because as we move up the g value decrease and t increase according to formila
As the elevator accelerates upward, the effective 'g' increases. Careful with the signs on g and a.
The overall G effective inc and dinominator increases and over T/G(eff) dec and (t/g+a)^1/2 dec and therefore time period decreases .
Amazingg!! ❤
Thank you! 😄
Wonderfully explained
I really like your videos. Thanks for these great videos.
I want to say that, Just moving upward will not give positive sign for “a”. It also needs to be speeding upward. If a lift moving upwards and slowing “a” will be negative.
You bring up an excellent point. It is important people not confuse upward velocity with upward acceleration.
Is this same for a accelerating train.i mean instead of the acceleration being upward it is to horizontal
Yes, except in the case of horizontal acceleration you need to use the pythagorean theorem to come up with the total acceleration.
Love the diagrams! Keep up the good work😍
Admittedly not the most complicated of sketches, but I'll happily take the accolade. Thanks!
Why would the acceleration not appear to increase when travelling at some constant velocity. The ball hits the floor faster if travelling upwards with some constant velocity.
The ball does not hit the floor faster when the elevator is travelling upward at constant velocity. Remember, if the person drops the ball while the elevator is moving upward, the person AND the ball are also moving upward. The ball would not instantly stop moving upward the moment the person lets go of the ball.
@@INTEGRALPHYSICS ah of course, thank you so much
Why would the x component of tension grow, if the acceleration is upward, the T grows but only in the Ty direction, no?
In order to increase Ty, T must increase. If T increases Tx must also increase.
Good video 👍🏻
Thanks 👍