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# Pendulum in an accelerating car

Dear Experts,

My question is, 'WHY exactly does a pendulum move to one side in an accelerating car(attached to the roof)'?
This is one of the most fundamental things that is needed to effectively solve Physics problems related to pseudo forces and inertial and non-inertial frames.

I hope you guys can help me out here. :)

Regards,
bk_jreinstein
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bk_jreinstein
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4 Solutions

Author Commented:
Oh, and i hope you can explain this to me using Newton's laws.
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Commented:
It's newton's first law

Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

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Author Commented:
How come it doesn't move with the same acceleration as the car thus keeping the string facing downwards? It is still theoritically attached to the car through the string.
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Commented:
when the string is vertical, then there is tension in the string exerting an upwards force on the ball, this force exactly counteracts gravity.

The vertical string cannot exert any lateral force on the weight though, for that to happen the string has to become angled.

As the weight is 'left behind' by the acceleration of the car, the string gets pushed back at an angle, this angle means that effectively some of the force of tension is acting in a lateral direction and some in an upwards direction.

If the string was fixed and rigid, (an iron bar welded to the roof say), then there would be something like the effect you envisaged (although the tension would still have to exist and cause some displacement, just no longer visible.
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Commented:
The whole idea of acceleration can get quite complicated, including reference to the General Theory of Relativity. But your question can be answered easily by Newtonian physical concepts.
Consider a frame of reference attached to the ground on which the car is moving. When the car accelerated the ball must also (otherwise it would blow out the back window).
Newtons 2nd law says the F = ma where F (the force on the object) and a (the acceleration of the object) are both vectors. In your case the net acceleration on the ball must be in the direction that the car is moving. That means that the net force must be in the same direction. Now let us look at the force on the ball. When the car is at rest there are two forces on the ball: the force due to gravity and the force due to the string. Both of these forces are equal in magnitude but in opposite directions. Therefore the net force is zero and the ball is not accelerated.
Now let the car accelerate forward. If the ball is to accelerate also there must be a net force on the ball in the forward direction. The force of gravity does not change, so the force due to the string must. The only way the string can exert a forward force on the ball is for the ball to be positioned behind the point at which the string is attached to the roof of the car.
The ball will take up that position which lets the string apply a horizontal force to the ball of just the required amount necessary to let the ball accelerate at the same acceleration of the car.
The force of gravity does not change, so the vertical component of the force on the ball due to the string must not change either (if the ball is to not move up or down). These relationships allow you to compute the angle that the string makes with the vertical, if you are given the acceleration of the car
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Commented:
The acceleration due to gravity is downward.
The earth/floor holds you up.
If you drop a ball, it will fall DOWN.

In a forward moving car, inertia holds you in place.
The car pushes you forward.
If you drop a ball it will fall DOWN and BACK.

The net force you FEEL is the sum of the gravitational and reaction vectors.
This direction of this new vector defines the local VERTICAL in the accelerating frame of reference.

Pendulums align themselves to this new vector.  The bob goes backward.

So will helium balloons.  It will go FORWARD.

Think about what bubbles would do.
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Author Commented:
Firstly, deighton, what exactly do you mean by weight being 'left behind'? I think you're missing my point. I know that it happens. I'm asking you WHY it happens. Also, please explain why the same does not happen with an iron bar.

Secondly, aburr, I don't want to know how to solve a problem relating to this. I am asking this question solely to attain a better understanding in this field. You've also not mentioned anything about why the ball should suspend itself at the angle. Why does it not accelerate with respect to the car itself?

Thirdly, d-glitch, inertia does not exist. What the common man calls inertia is actually nothing but Newton's first law. Please tell me why only the upper body experiences this force and not the whole body.
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Commented:
when the car starts to move, there is initially no force acting on the pendulum weight in a forwards motion to start it moving with the car, so initially it simply stays where it is relative to the road below.

in the case of an ideally rigid iron bar, the weight has to either move with the bar or snap off

>>>also not mentioned anything about why the ball should suspend itself at the angle.

To pull an object along a surface with a string, the string can not be 90 degrees vertical to the object, otherwise it can only lift it upwards.

The angle produces a state where the pendulum weight is having a force applied to it vertically to cancel gravity, and laterally to accelerate with the car.
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Commented:
"aburr, I don't want to know how to solve a problem relating to this. I am asking this question solely to attain a better understanding in this field. You've also not mentioned anything about why the ball should suspend itself at the angle.
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If you do not want to solve problems, don't.
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" You've also not mentioned anything about why the ball should suspend itself at the angle."

Done in the sentence below copied from my post.

If the ball is to accelerate also there must be a net force on the ball in the forward direction. The force of gravity does not change, so the force due to the string must. The only way the string can exert a forward force on the ball is for the ball to be positioned behind the point at which the string is attached to the roof of the car.
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Newton's second law (quoted in my post) says
F = ma.
Thus it says that the acceleration points in the same direction as the force and is directly proportional to it.
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Commented:
Imagine a stationary car, with the pendulum in it, and the car starts accelerating as rapidly as possible.  As the car first moves, that very first bit of acceleration, there is nothing to put a force on the pendulum weight to set it moving, just as a bird hovering above the car would retain its position relative to the road, so would the pendulum weight.

Now the top of the pendulum attached to to the roof, that does move with the car, otherwise it would have to break off.  So therefore there has to be an inclination of the pendulum string starting to appear, and it is this inclination of the string that passes on the force to the pendulum weight.
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Commented:
>>>Please tell me why only the upper body experiences this force and not the whole body.

that's not generally true, it's more to do with rigidity and freedom of movement.  If you assume a car structure is 100% rigid and the top of the pendulum fastening is 100% rigid, then this part has to accelerate with the car, but the weight is hanging by string which is flexible, so the weight is capable of initially staying at rest.

more examples

If you stand on top of a car covered in ice, and the car accelerates away super quickly, then the ice can apply no lateral force to your feet, the car goes and you are left in space exactly where you were, until gravity takes over.  (newtons first law).

Look at it the other way, unbelted man in car going 50mph in a car, hits brick wall, car stops dead, there's nothing to stop the man moving, he keeps moving at 50mph until he himself hits something to stop him.
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Commented:
It is not surprising that you do not have an intuitive feeling for what is happening.
For thousands of years people thought that experiment showed that Newton's law could not be true. A better understanding of friction (and other experiments) showed that Newton's ideas represented a better way of looking at nature.
You cannot develop an intuitive feeling without a lot of effort. Presumably you have done the experiment. You can find no simpler explanation of the experiment than has been mentioned here.
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Commented:
Minor note... The effect on the pendulum is also an effect on the roof of the car, the seats, the steering wheel and even the tops of the tires and wheels. EVERY part of the car lags behind as force is applied at the friction points to move the "car" forward.

It's not so obvious that the roof also lags behind because the rigidity of the frame and body minimize the effect to the point of it not being visible to the human eye. But the effect exists and can be measured with enough precision.

Tom
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Commented:
Isaac Newton was around 300 years ago, so there wasn't thousands of years to spend doubting his laws.
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Commented:
Not being particularly well versed in physics myself, I read this post with interest in learning more and have struggled to comprehend the earlier, more technical posts.  However I think I understand the general concept now, but to clarify what i've learnt to check it's correct:-

When the car starts to accelerate, all objects attached to it will also begin to accelerate, but at slightly delayed times based on their position to a fixed point in/on the car.  Therefore the top of the string will accelerate before the bottom of the string but the whole pendulum will eventually reach the same speed fixing it at an angle.
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Commented:
Note that the angle of the pendulum only exists due to and during acceleration, if the car reaches a fixed speed, then the pendulum goes back to being vertical.

Same as you would feel pushed back into your seat during rapid acceleration, but once at cruising speed this would cease (note though that the feeling of being pushed pack into your seat is in fact the seat pushing you forward to accelerate you with the car).

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Commented:
Isaac Newton was around 300 years ago, so there wasn't thousands of years to spend doubting his laws.
I know very well when Newton was born and when he proposed his laws. The point was that for thousands of years experiments (formal and informal) were interpreted as showing that a force was necessary to keep a body in uniform motion which is contrary to Newtons vastly simpler and revolutionary ideas.
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Commented:
When the car starts to accelerate, all objects attached to it will also begin to accelerate, but at slightly delayed times based on their position to a fixed point in/on the car.  Therefore the top of the string will accelerate before the bottom of the string but the whole pendulum will eventually reach the same speed fixing it at an angle.
Not quite. All points rigidly attached to the car will accelerate exactly and just as soon as the car does. The kicker here is rigidly. Most of the car is sufficiently rigid that you can consider the car to be rigid for all practical purposes.
HOWEVER the pendulum is NOT rigidly fastened to the car. The string (when vertical) cannot transmit the horizontal force to the bob which Newtons law requires to give it a forward acceleration. Only when the string is at an angle can a component of the force that the string applies to the bob provide the bob with the forward acceleration necessary to keep up with the car.
Note that when the car has reached its maximum velocity and is no longer accelerating the string will once again be vertical. At this time the bob sill be no longer accelerating in a horizontal direction so there is no longer a need to have a horizontal force on the bob.
When the car starts to accelerate, all objects attached to it will also begin to accelerate, but at slightly delayed times based on their position to a fixed point in/on the car.  Therefore the top of the string will accelerate before the bottom of the string but the whole pendulum will eventually reach the same speed fixing it at an angle.
Not quite. All points rigidly attached to the car will accelerate exactly and just as soon as the car does. The kicker here is rigidly. Most of the car is sufficiently rigid that you can consider the car to be rigid for all practical purposes.
HOWEVER the pendulum is NOT rigidly fastened to the car. The string (when vertical) cannot transmit the horizontal force to the bob which Newtons law requires to give it a forward acceleration. Only when the string is at an angle can a component of the force that the string applies to the bob provide the bob with the forward acceleration necessary to keep up with the car.
Note that when the car has reached its maximum velocity and is no longer accelerating the string will once again be vertical. At this time the bob will be no longer accelerating in a horizontal direction so there is no longer a need to have a horizontal force on the bob.
deighton has said as much in a more intuitive fashion.

I said above that The string (when vertical) cannot transmit the horizontal force to the bob which Newtons law requires to give it a forward acceleration. That is not quite true. Newtons Law does not require that the bob have a horizontal force on it. The universe requires it. Newton just identified and quantified this requirement.
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Commented:
>> When the car starts to accelerate, all objects attached to it will also begin to accelerate, but at slightly delayed times based on their position to a fixed point in/on the car.  Therefore the top of the string will accelerate before the bottom of the string but the whole pendulum will eventually reach the same speed fixing it at an angle.
> Not quite.

Yes, quite.

Simultaneity really doesn't happen, like it or not. AFAIK, we don't even have theoretical materials that would transmit force necessary for "exactly and just as soon" acceleration through the entire car. I'm not sure we even have viable theories that would allow someone to propose such a theoretical material.

Accelerate one end of a rod held perpendicular to the direction of acceleration and you will see the rod bend, as long as the rod is long enough for it to be visible. The entire rigid rod does not accelerate "exactly and just as soon" as the point where force is applied.

Even when the rod is too short for the effect to be visible, it still happens. An inability to see it does not equate to 'does not happen'.

With precise enough equipment, the deformations can be made visible. Over time, the deformations can cause catastrophic failures due to metal fatigue effects, etc. (Extremely unlikely in a car, of course.)

Tom
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Commented:
"Simultaneity really doesn't happen, like it or not. AFAIK, we don't even have theoretical materials that would transmit force necessary for "exactly and just as soon" acceleration through the entire car. I'm not sure we even have viable theories that would allow someone to propose such a theoretical material."

Indeed you are correct. We do not know exactly what is going on in the universe (and probably never will). The special theory of relativity and the general theory have proven that Newton's laws are not exactly correct.
However the original question was asked in the context of Newton's Laws by people who were having a difficult time understanding the problem at that level. It is useless and indeed counter productive to insist that people understand tensors before one understands something about the simpler relationships between acceleration, force, motion, and time.
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Commented:
> Newton's Laws

...apply just fine for this example. The car does flex. It doesn't have to be considered in relativistic terms. The "rigid" material is merely 'more rigid' than the string. If the flexing of the car body can't be brought into the concept, then there's no hope of explaining why the string flexes and the pendulum bob goes out of perpendicular.

The string may be attached via 'frictionless swivel' or attached directly. Does it matter much in terms of the question? No, because there will be no visible difference. If it's a fixed attachment, then the string will bend just as the car body/frame bends.

Due to basic laws of intertia.

A 'rigid' skyscraper would illustrate the concept during an earthquake. Acceleration is applied at the base as the ground moves. The top of the building does _not_ begin moving at the same time. The building bends, potentially enough to fracture concrete, masonry and glass.

Not from relativity (in the macro realm). There is no need even to think about relativity in this thread.

Standing still, a tall building can seem to be _very_ rigid. Fine. Just don't try to convince engineers that it really really is.

Recognizing and understanding how the building reacts (or the material of a car body/frame) takes the same principles as the pendulum. To try to define the pendulum as a true 'special case' simply provides a false description that doesn't fit with the rest of the world.

If it's a very lightweight string, it provides very little resistance to the tendency to stay motionless. Replace it with a thin wire, and resistance (called 'rigidity' here) increases; but a flexible enough wire will approximate a string. Replace the wire with a thin rod (to differentiate from 'wire'), and the resistance increases even more.

Replace it with an I-beam, and 'rigidity' becomes practically infinite (for practical purposes here). But it really really isn't infinite. It really really still has a pendulum effect. As does the body of the car. Various accelerometers use that "fact" specifically to measure acceleration.

If imagination is so limited that it's a struggle to extend the conceptual image of a pendulum to material in general, then I see no hope in expecting 'intuitive' results.

It was failures of imagination that kept science from seeing that a falling apple was the same as a 'falling moon in orbit'. We should have gotten past that point after Sir Newton told us otherwise. To me, it's counter-productive to insist that people can't simply imagine metal bending when force is applied.

To me, once it's accepted that it happens, then the pendulum becomes an obvious consequence. No fancy math needed.

Tom
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Commented:
I agree that none of the car can be ideally rigid, but a good model to use, both towards understanding and computation, is that the car itself is ideally rigid and can switch to a state of continuous acceleration instantaneously.
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Author Commented:
So whether or not a string/rod bends depends on its rigidity?? Sorry for the late follow up.
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Commented:
"So whether or not a string/rod bends depends on its rigidity?? Sorry for the late follow up."
Yes. But that point has nothing to do with the background of your question.
In any physics problem simplifications are made. Your question is a classic general physics problem. As such, strings have no rigidity. Any time lag between application of a force to a rigid body and its result is zero.
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"WHY exactly does a pendulum move to one side in an accelerating car"
The answer is that it does not. That is, it does not move to one side in the frame of reference which should be used in this problem. That frame is the ground. In that frame the bob does not move. The car moves and a force is needed to make the bob accelerate with the car. To apply that force through the non-rigid string requires that it pull on the bob at an angle.
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Commented:
if the pendulum and the car are moving North, and a force deflects the car to the East, the pendulum will continue to move North according to Newtons Laws, until a force also accelerates it to the East.
From the car's point of view of a car going East, a pendulum continuing on its previous path will be going West.
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Commented:
I think what you maybe need to understand is that being inside the car isn't in itself a reason for the bob to move with the car.  The fact that air in the car (appears) to move with the car wouldn't be enough to move the bob, so it has to be the string that accelerates the bob.  The string has to be at an angle to produce the necessary force in the direction of acceleration.  The string acquires this position by virtue of the fact that the bob does not initially move

Rigid substances apply force by undergoing very small amounts of flexing, that you might not be able to see.

Another thing to contemplate is that thinking of a pendulum in a static situation, swinging back and forwards, how fast is the ball traveling at the top of its swing before it goes back the other way?  the answer is its velocity is zero, at the highest point of its swing its come to a halt.  But its acceleration is not zero, this point is where acceleration of the bob is maximum.
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Author Commented:
Im sorry, i just allocated to points your last posts and not necessarily the solution posts...
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