# Newton's $1$st law question

A long time since I posted it, still did not get a satisfactory solution. So I restate the problem -

Why does not an object attached to a string which is attached to the roof of an accelerating bus does not stay in it's initial position for an indefinite amount of time(and causing the string to stretch so much so as to break it)? In other words what is the time for which the object stays in its initial position? I expect it to stay stationary at that point, since it does initially, when a sudden change in velocity occurs.

One can still see unedited form of the question from the question history (You might want to give it a look, since it differs significantly, but the question is quite the same).

• Are you done editing? Mar 6 at 16:13
• @BobD What else to do when your question gain no attention? Mar 6 at 16:14
• I understand. But every time I am about to post an answer your post gets edited. Very difficult to shoot at a moving target, that's all. Mar 6 at 16:17
• It sounds to me like you are asking: "Why doesn't the object stay at a fixed location with respect to the road." Is that what you are asking? "I expect it to stay stationary at that point, since it does initially, when a sudden change in velocity occurs." There is no sudden change in velocity. No sudden anything. Mar 6 at 19:01
• @Eisenstein When you say "initial position", do you mean with respect to the coordinates of the bus or with respect to the coordinates of the road? Mar 10 at 12:12

Assume the object hangs vertically before the bus starts to accelerate. When the bus starts to accelerate there is instantaneous relative movement of the object relative to the bus seen by observers both on the bus and on the ground. To the observer on the bus, the bus is not moving and the object moves due to the fictitious force present in the accelerating (non-inertial) reference frame attached to the bus. To the observer on the ground (inertial reference frame), the object is stationary and the bus moves.

In both cases the relative motion of the object relative to the bus is the same and the object continues to move relative to the bus until a component of the tension force (a) counters the fictitious force for the observer on the bus and, equivalently, (b) equals the acceleration of the bus for the observer on the ground. The object moves until it reaches a new equilibrium position. If the object is "tapped" (slightly displaced from equilibrium), it acts as a pendulum about the equilibrium position. See Time period of a simple pendulum in an accelerated frame.

What is the value of acceleration when the string breaks?

The string will break when the tension in string reaches the tensile limit say ($$T_0$$).The tension in string for given acceleration can be calculated easily from FBDs.

1. Why does not the ball stays fixed in its initial position forever, so that the tension becomes high in the string, and it eventually breaks?

It depends on weight of the object.

Think of a case when the object is so heavy that the string is barely able to hold the ball and could break anytime . Even small acceleration in the bus would break the string.

• Why would I even THINK about such a special case? My question was about why cannot the object (irrespective of mass) just retain it's initial position, and stretch the string to such an extent that it breaks (1st law of Newton, inertia of rest)? Mar 6 at 13:23
• @Eisenstein You're asking why things can be moved at all from some ordained position.
– jpf
Mar 6 at 13:34
• @jpf In a nutshell yup. Mar 6 at 13:36
• @jpf But not the same question as this, since it involves the 3rd law of motion, while mine involves the first law. Mar 6 at 13:37
• @Eisenstein Since the first law only requires an external force applied, you must be asking from where the force on the object comes? It comes from the string, attached to the bus, which is propelling itself forward by way of forces on the tires from the road.
– jpf
Mar 6 at 13:54

Why does not an object attached to a string which is attached to the roof of an accelerating bus does not stay in it's initial position for an indefinite amount of time...

As pointed out by @John Darby, there is theoretically instantaneous movement of the object relative to the bus as seen by both the inertial and non-inertial observers as soon as the bus accelerates. That's because, per Newton's first law, a body at rest stays at rest unless acted upon by a net external force. The horizontal component of the string tension is that net horizontal force. See the free body diagram of Fig 1 below.

The angle $$\theta$$ between the string and the vertical axis of the coordinate systems of both frames is a function of the acceleration of the bus according to

$$\tan \theta =\frac{a}{g}$$

The tension in the string, as a function of $$\theta$$ and the mass of the object, is then

$$T=\frac{mg}{\cos \theta}$$

In the frame of the road the object falls behind the bus as the acceleration of bus increases until the acceleration and $$\theta$$ is a maximum. At which time the tension in the string is a maximum. See the sequence in Fig 2.

...causing the string to stretch so much so as to break it

The maximum tension in the string is limited by the mass attached to the string and the maximum angle $$\theta$$, which is a function of the acceleration of the bus. The maximum possible acceleration of the bus is limited by the coefficient of static friction between the tires and the road, according to

$$a_{max}=\mu_{s}g$$

Substituting into the first equation

$$\tan\theta_{max}=\mu_s$$

$$\theta _{max}=\tan^{-1}\mu_s$$

To put some numbers on this, the coefficient of static friction between the tires and dry pavement is generally in the range of 0.5 - 0.8. Taking the maximum value for static friction, $$\theta_{max}$$ = 38.7 $$^o$$. The maximum tension in the string is then

$$T_{max}=\frac{mg}{\cos 38.7^o}$$

One manufacturer of nominal 1/8 in (3.2 mm) diameter sisal twine says its breaking strength is about 1500 N. Using this in the previous equation we can calculate the mass of the object needed to break this string. That mass would be about 119 kg.

Hope this helps.  • It is actually a great answer! Thanks. Mar 11 at 6:16

The answer is quite simple, the string attached to the roof pulls the object with it. Now there is a maximum force that the string can withstand, its called the tensile strength, beyond which it simply breaks.

• Actually not that simple. Everybody knows the fact you posed. My question was actually different from that. It does not happen so that the object remains stationary until the string breaks; the object moves with the string sometimes later. That was what my question was all about. Mar 6 at 16:33
• @Eisenstein You need to tell us exactly what you mean by "sometimes later". I posted then deleted an answer when I saw this comment. Does it have something to do with whether or not the string is extensible? Mar 6 at 16:54
• @BobD No. When I say sometime later, I literally mean the action to the string and the object which takes place when you suddenly accelerate bus. I can elaborate if you wish. Mar 6 at 16:58
• @Eisenstein I think you have to elaborate because I, for one, am unable to follow you. Mar 6 at 16:59
• @BobD Think of what happens when you accelerate the bus. The object tries to retain it's initial position (inertia of rest), but the part of the string attached with the roof moves, causing the object to leave it's initial position. My question is why doesn't the object remain permanently in it's initial position? I hope I am clear. Mar 6 at 17:03

*@Eisenstein You're asking why things can be moved at all from some ordained position. – jpf

@jpf In a nutshell yup. – Eisenstein*

It seems physics has offered the best answer that it can with regard to a questions of attaching a number to the concept we call inertia (the proportionality constant we call mass that results in an equality in a force and acceleration relationship).

The inertia concept is an observation that we make about events we see in our environment. Newton tried to make sense of these and created the useful fiction of a fixed background of time and space. He then went on to say matter is referenced with respect to a "position" at some "time" in this useful fiction. It worked and gives us useful answers as long as we can pretend that someone can act as an observer standing outside the experiment referenced to this framework.

Of course this perspective starts to fall apart as soon as we start to think of cosmology and wonder why the universe is not 5 meters to the left instead of where it is now.

So I think you are asking a very deep question about "an ordained position at some point in time" at a level that we cannot really answer with our present state of knowledge. There are physicists who do think this is an important question and that ti might be tied up with the basic problem of quantum gravity and our lack of progress with it, as well as other paradoxes like a small point-like start to the universe and now some massive "volume" to the universe. Point and volume in what framework? We can't stand outside the universe.

Hopefully this makes some sense and you can appreciate the difficulty in answering what seems like a simple question about an object of matter at some "ordained position" in a deeper way than can be answered using Newton's convenient fiction of space and time.

• My question is not at all as much complicated as you make it look! Mar 6 at 14:40
• I will just point to the initial quote I referenced. Mar 6 at 14:52

According to relativity, the mass (which is the quantitative analogy of inertia: the higher an object's mass, the harder it is to accelerate it under the influence of a force) attached to a spring in an accelerated frame is equivalent to a spring hanging in a gravity field. Objects in this gravity field will experience the same acceleration as the acceleration of the frame in free space. Einstein used the example of an accelerated elevator in empty space. Say we attach a spring on the inner top of the elevator (which becomes the top only when accelerated). We give the elevator an acceleration. Like in the bus, the mass will have the tendency to stay behind (inertia, mass). It will not stay put though. Only when the spring can provide enough force the mass will stay put relative to the elevator (bus).
Someone inside the elevator can describe the situation just as well by saying he finds himself in a gravity field (if he can't look outside). If we find ourselves in a gravity field, say standing on a beach, we are accelerated upward too, just as in the elevator or bus. I think you don't any difficulties with a mass hanging on a spring in your backyard.

Note that gravity is not a real force. It's an inertial force. What we feel when standing on the ground is the electromagnetic force pushing us up. There is no inertial force without the electromagnetic interaction (or the strong or weak interaction, though it's kinda hard to use them).

Right now I see that you meant a string instead of a spring. And that the mass is hanging in a direction perpendicular to the acceleration of the bus. This doesn't alter the reasoning.

If the mass would stay in its original position, why won't the bus stay in its original position when trying to accelerate it?

• But this does not answer my queries! Mar 6 at 13:49
• If you know how a mass that you hang on a spring behaves then you know how a mass attached to a spring in an accelerated bus behaves. The two situations are equivalent. Mar 6 at 14:20
• Nope, they aren't equivalent. One is inertial, while the other is accelerating. Mar 6 at 14:24
• And spring makes life more complicated! Mar 6 at 14:25
• It's almost spring! The mass in the gravity field finds itself in an accelerating frame. So do you when standing on the floor. The floor accelerates upward. Both masses experience an acceleration. Mar 6 at 14:42