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Suppose a pendulum bob is hanging to the ceiling of the car. We know that when we accelerate the card forwards then the pendulum bob moves backwards. But why does it move backwards? If it moves backwards there must be some force acting on the pendulum bob. But what force is acting on the bob? I know that if we look at the pendulum bob from the car's frame of reference or from the non-inertial frame of reference then we may say that the pseudo force acts on the bob due to which the bob moves backwards. But pseudo force is not a real force. It is a fictitious force or rather a correction factor which we use when we are using Newton's laws of motion from a non-inertial frame of reference because Newton's laws of motion and not valid in non-inertial frame of reference. So what force actually acts on the pendulum bob which causes the bob to moves backwards? If we are looking from the inertial frame of reference or if we are looking from outside the car then we would see the pendulum bob move backwards but what force acts on the body that makes it move backwards? Who applies this force on the bob? Moreover if we imagine to be the bob itself then we would feel a real force pushing us backwards (for example, when we go around a merry-go-round we obviously feel a real force that seems to be pushing us radially outwards or when we are sitting on a card and the car suddenly accelerates we seem to feel a force that is pushing us backwards; what exactly is this force and who is applying it on us?) (Also, in theory, we say that pseudo force is a fictitious force but doesn't it seem to be a real force? I mean when we go around the merry-go-round we obviously feel a real force that is pushing backwards, right? So doesn't pseudo force feel more like a real force that acts on us?) Can someone please answer why the bob moves backwards? More specifically, what force acts on it which keeps it backwards?

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    $\begingroup$ The bob is not going backwards in the rest frame of the street around you. $\endgroup$
    – my2cts
    Jun 16, 2021 at 10:11
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    $\begingroup$ Well you need to understand the first law of motion. Secondly relativistic mechanics is accounted for particles moving with speeds comparable to that of light. In the case of pendulum bob it is inertia that is showing reluctance to move from its stationary position. $\endgroup$
    – Stack3002
    Jun 16, 2021 at 10:17
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    $\begingroup$ Why is the station going backward when the train starts to roll? $\endgroup$
    – my2cts
    Jun 16, 2021 at 10:24
  • $\begingroup$ For even more fun, repeat this setup but with a helium balloon tied to the seat (and floating above it) $\endgroup$ Jun 16, 2021 at 12:56
  • $\begingroup$ Re, "But pseudo force is not a real force." It's real in the frame that accelerates with the car. "Real" in the sense that, you can't mathematically describe the motion of the pendulum in that frame of reference without invoking the pseudo force. But, we call it "pseudo" because it does not exist (i.e., it is not needed to explain or describe anything) when you look at the same scene from the perspective of an un-accelerated frame. $\endgroup$ Jun 16, 2021 at 13:34

4 Answers 4

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The 'real' forces acting on the pendulum bob are 1) the tension in the pendulum string and 2) the gravitational force. If we're looking from outside the car, the sum of these two forces is the source of the bob's acceleration. So Newton's equation of motion for this bob is given as follows: \begin{equation}\vec{T}+m\vec{g}=m\vec{a}. \end{equation}

Note that when looked from outside, not only the car but also the bob is accelerating forward. We can also observe that if the bob goes forward, the net force points backward, and this is in contradiction to the assumption that the car is accelerating in the forward direction.

In the (non-inertial) frame of reference moving along with the car, Newton's equation of motion for the bob is given as follows: \begin{equation}\vec{T}+m\vec{g}+(-m\vec{a}) = \vec{0}. \end{equation} By introducing the fictitious force $(-m\vec{a})$, we are successfully able to explain the fact that the bob is not accelerating (and not moving) in this frame. In this sense, we can think of this fictitious force as a physical tool introduced to make Newton's equation of motion valid even in the non-inertial reference frames.

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As viewed from someone sat in the car next to it the bob appears to go backwards - but what's really happened is that the roof of the car and the person watching move forwards and the bob stays where it is, initially.

It'll only appear to keep the 'backwards' position if the car is accelerating.

If the car were to brake, since the person and roof of the car slow down first and the bob keeps going at constant speed, it would appear to move forwards and stay in a forward position as long as the car continues to brake.

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Suppose a pendulum bob is hanging to the ceiling of the car. We know that when we accelerate the card forwards then the pendulum bob moves backwards.

The bob only appears to initially move backwards to the observer in the non inertial (accelerating) reference frame of the car. It appears to accelerate backwards because the observer is accelerating forwards. As you already appear to know, in order to apply Newton's laws of motion in the non inertial frame the apparent backwards acceleration is attributed to a pseudo (fictitious) force,

But to an inertial (non accelerating) observer on the road the bob initially appears to remain in place. Eventually to the observer on the road the bob moves forward together with the car, whereas to the observer in the car the bob eventually remains in place. This is discussed further below.

If we are looking from the inertial frame of reference or if we are looking from outside the car then we would see the pendulum bob move backwards but what force acts on the body that makes it move backwards?

That is not correct. To the observer in the inertial frame of the road the bob does not appear to move backwards. It initially appears to remain in place while the vehicle moves forward.

The explanation is Newton's first law which states that an object at rest tends to remain at rest unless acted upon by a net external force. It's a statement concerning the inertia of an object. The net external horizontal force acting on the bob (neglecting air resistance) is the horizontal component of the tension in the string. That component is initially zero and builds as the vehicle accelerates.

The remaining questions of your post are difficult to answer point by point because the they are interrelated. So instead consider the following explanation of the motion of the bob in the inertial frame of the road.

See the figures below.

In the top figure I have the pendulum secured to the top of the roof.

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).

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 increasing horizontal component of the tension force in the string. Eventually the pendulum angle will reach a maximum value, depending on the car's acceleration, and remain at the angle as long as the car continues to accelerate. At this point the horizontal velocity of the bob will be the same as that of the car.

At this state, to the observer in the car, the bob will now appear stationary at a fixed angle relative to the vertical, again as long as the vehicle continues to accelerate.

When the vehicle's velocity becomes constant, the bob will hang vertically to both observers. To the observer on the road the bob moves at constant velocity with the car, whereas to the observer in the car the bob is not moving at all.

Hope this helps.

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Heres the intuition part:

(For the whole solution and boring details, you can see my answer here: https://physics.stackexchange.com/a/653596/307354 )

INERTIAL FRAMES INTUITUON

How does this force work inside the inertia frame? I always remember that if a room is falling, it is accelerating downward and has no overall force inside the room. Like zero gravity. So acceleration down makes apparent force up, which cancels the gravity force down. Accelerating down at the acceleration of gravity provides an upward-seeming force equal to the force of gravity. That’s why the inertial acceleration force is -a.

INERTIAL FRAMES STRATEGY

The key is get the one apparent gravity (by adding the vectors of g and -a) and forget youre moving and forget which part is actual gravity and which part is inertial, and then work on the problem.

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