Consider this following thought experiment: There are two tracks, one with a line of cameras and another on which a car with transparent sides moves (*). Now, imagine attaching a pendulum to the roof of the car, as the car accelerates to gain speed, we see that the pendulum makes an angle with the vertical in a direction opposite to which the car is accelerating in from the photographs which the line of cameras will take.

Using the photographs, we can construct a position time curve and we would find that the acceleration of bob in the car has some sudden shifts. A person inside the car can understand this phenomenon as the effect of pseudo force(**), that is the pendulum bob experiences a force opposite to the direction in which the car is accelerating. However, how does a person outside the car rationalize this phenomenon occurring?

*: We can think of it as ripping off the car door so that we can see the pendulum's motion inside the car at all times.

**: If we attach a frame of reference to the car, then that frame is accelerating, and hence we have to consider the -Ma pseudo force on it. Where M is the mass of the pendulum and a is the acceleration of the car.


3 Answers 3


However, how does a person outside the car rationalize this phenomenon occurring?

When the car initially accelerates, the person outside the non inertial frame of the car sees the pendulum bob as "staying in place" while the car accelerates forward, i.e., the bob does not accelerate with the car. The person explains it as due to the inertia of the bob as a consequence of Newton's first law, namely, that a body at rest tends to stay at rest unless acted upon by a net external force.

The person inside the car can only explain the initial backward acceleration, based on Newton's second law, as being due to some force acting on the bob, the same force that pushes the person back against the seat when accelerating. Since there is no contact force acting on the bob or pushing the person back against the seat, it is considered a pseudo or fictitious force.

Once the car accelerates and maintains its acceleration, the bob will stay in place in some backward position and actually experience a real acceleration along with the car due to the tension in the bob string.

"When the car initially accelerates, the person outside the non inertial frame of the car sees the pendulum bob as "staying in place" while the car accelerates forward, i.e., the bob does not accelerate with the car. " Doesn't the air molecules inside the car bouncing around have any effect on the motion of the Bob? I mean would the result be same even if we did this in a space ship in vacuum?

If the bob had motion due to the air, that would mean it is being influenced by contact forces with the air molecules. Remember, Newton's 1st law states that an object at rest tends to stay at rest unless acted upon by an external force. If motion of the bob were due to contact with air molecules, that would constitute such an external force. Naturally, in the vacuum of space no contact with air molecules would be involved.

I think the confusion I'm having is the part about the pendulum staying in place even when the car starts moving. It is like we can clearly see the angle which the pendulum rope makes with the vertical axis changing. So, clearly, the configuration of the system is changing yet we say that the pendulum is staying in place

The pendulum bob only initially stays in place horizontally in the frame of the road due to no external horizontal force. But as the horizontal component of the string tension increases the bob moves with the car.

The diagrams below may help to visualize what's happening. The first digram shows the pendulum attached to the roof of the car from the perspective of an observer in the inertial frame (the road). We will assume no air drag force will arise. So initially there are no external horizontal forces acting on the bob.

Figures A-C of the second diagram show the progression of motion of the car/pendulum as seen by an observer on the road (the non accelerating frame). The acceleration begins at Fig A.

In Fig B the car/pendulum base has moved in the x-direction. The bob tends to stay in place horizontally due to its inertia with only a slight displacement (less than the car/base) in the x-direction due to a small horizontal component of the tension in the string.

In Fig C the bob is increasingly being pulled by the string in the x-direction due to the force in the string. Eventually the pendulum angle will reach a maximum value, depending on the car's acceleration, and the horizontal velocity of the bob will be the same as the car.

Hope this helps.

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  • $\begingroup$ If the train has been accelerating for a while, the the person outside sees the bob being accelerated with the train by the horizontal component of the tension in the support. $\endgroup$
    – R.W. Bird
    Oct 21, 2020 at 14:01
  • $\begingroup$ @R.W.Bird Yes, because then the real centripetal force in the string will be acting on the bob to cause it to accelerate with the car. My answer relates to the initial observation of the bob "appearing" to move back in the car (non inertial frame) as opposed to appearing to remain in place in the inertial frame. $\endgroup$
    – Bob D
    Oct 21, 2020 at 14:09
  • $\begingroup$ @R.W.Bird But I will qualify the difference between the initial effect of acceleration and the continuous effect of acceleration. $\endgroup$
    – Bob D
    Oct 21, 2020 at 14:25
  • $\begingroup$ If the mass of the pendulum isn't entirely in the bob, the bob will initially accelerate backwards. $\endgroup$
    – benrg
    Oct 21, 2020 at 16:41
  • 1
    $\begingroup$ @Buraian I assumed the base of the pendulum is permanently attached to the roof of the car because you stated "imagine attaching a pendulum to the roof of the car". What else can "attaching" mean unless the base is somehow affixed to the roof of the car (bolted to, glued to, or held by static friction). If it were not attached the whole pendulum would stay in place in the inertial frame while the car accelerates. $\endgroup$
    – Bob D
    Oct 23, 2020 at 16:11

I think the best answer here is that, from the perspective of the inertial observer, there just isn't anything to explain, as long as you've accepted Newton's laws. The pendulum bob is doing exactly what Newton's first law predicts it will do: maintaining its momentum in the absence of any forces on it.

The need for explanation arises only for the non-inertial observer, who sees a violation of Newton's 2nd law insofar as he sees the bob accelerate but with no force to explain it. So he has two options: either invent a force to explain the acceleration or postulate that his reference frame itself is accelerating. This is analogous to how Einstein revolutionized our understanding of gravity.


The person outside the car can understand the pendulum's behavior in the same way as the person inside. Any reference frame can be used to solve any problem, so you should pick the one that makes the problem easiest. You aren't obliged to pick a frame in which you're at rest. When people do, it's only because that makes the problem easier—and it certainly won't here, given that the people have nothing to do with the problem setup.

I think that modeling the pendulum's behavior in this problem, if you want to be fairly accurate about it and include oscillation and perhaps damping, will be far easier in the accelerating frame, at least if the acceleration is uniform. You can take the sum of the gravitational and fictitious acceleration as your effective gravitational acceleration, and plug it into standard pendulum formulas. The description with respect to the inertial frame, however you derive it, will be a coordinate transformation of the simpler description with respect to the accelerating frame.

If you want a qualitative description with respect to the inertial frame, neglecting details of the motion, then you can read Bob D's answer. I realize this is a non-answer, but I feel like it's the correct response to word problems like this that appear, as they're phrased, to be based on a misconception. The last thing the world needs right now is the notion that different people live in private universes and are unable to comprehend anything from anyone else's point of view. I can tell you that that's 100% untrue in physics. You can understand the banding of Jupiter in terms of the Coriolis force even if you aren't on Jupiter, and you probably should.

  • $\begingroup$ The question isn't asking if one frame is better than another frame. The question is asking how one describes something from a particular frame. Saying that you can use any frame you want doesn't answer the question because the OP is asking about a specific frame. $\endgroup$ Oct 21, 2020 at 16:42
  • $\begingroup$ @BioPhysicist But the real-life answer to how you describe things in that frame is that you don't; you do things with respect to the most convenient frame. $\endgroup$
    – benrg
    Oct 21, 2020 at 16:44
  • $\begingroup$ Then you aren't answering the OP's question. They aren't asking whether or not to use the most convenient frame. That would be like someone wanting to solve a problem specifically using forces and Newton's laws and you saying, "well actually here it is better to use energy conservation". Sure there might be a better way to look at it, but they asked for a specific thing. $\endgroup$ Oct 21, 2020 at 16:47
  • $\begingroup$ @BioPhysicist There's a lot of discussion on meta about what to do when someone asks the wrong question. My answer says that it's the wrong question. Other answers answer the question as asked. I'm not suggesting the other answers be deleted; I even cited one. $\endgroup$
    – benrg
    Oct 21, 2020 at 16:59

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