# Circular Motion: What's the Source of Centripetal force in this?

• vinci
In summary, the question discusses the circular motion of a truck on a roller-coaster loop and the forces acting on the truck. The truck is kept in circular motion by a combination of gravity and inertia, with the net sum of these two forces providing the centripetal force. At the top of the loop, there is no normal force and gravity is the only force acting on the truck, causing it to be in free fall. The question also asks for the speed of the truck at the top of the loop, which can be calculated using the acceleration due to gravity.
vinci
[Moderator's Note: Thread moved from forum General Physics hence no formatting template]

I am trying to study Circular Motion for my exams and I'm kind of unsure about one question. The question asks what's keeping the truck in circular motion. It has to be gravity I know, but gravity being directed towards the center, shouldn't that just result in the truck falling? What keeps it INTACT to the track? I am giong to quote the actual question now.

"Figure 18.17 shows part of the track of a roller-coaster ride in which a truck loops the loop. When the truck is at the position shown there is no reaction force between the wheels of the truck and the track. The diameter of the loop in the track is 8.0 m.

a) Explain what provides the centripetal force to keep the truck moving in a circle.
b) Given that the acceleration due to gravity g is 9.8 m s-2, calculate the speed of the truck.

"

Why gravity?

What keeps the truck moving in a circle when it's at the 3 o'clock or 9 o'clock position? What about at the 5 and 7 o'clock positions? What other force(s) are acting on the truck - in particular, pushing it towards the centre of the circle (which is what centripetal means)?

vinci said:
I am trying to study Circular Motion for my exams and I'm kind of unsure about one question. The question asks what's keeping the truck in circular motion. It has to be gravity I know, but gravity being directed towards the center, shouldn't that just result in the truck falling? What keeps it INTACT to the track? I am giong to quote the actual question now.

"Figure 18.17 shows part of the track of a roller-coaster ride in which a truck loops the loop. When the truck is at the position shown there is no reaction force between the wheels of the truck and the track. The diameter of the loop in the track is 8.0 m.

a) Explain what provides the centripetal force to keep the truck moving in a circle.
b) Given that the acceleration due to gravity g is 9.8 m s-2, calculate the speed of the truck.

"
It is gravity and, as you conclude, at this point (for an instant) the truck is in free fall. To see what's happening, imagine the track was only the right half of the loop, so that after the highest point, there is no track to the left. What would happen to the truck?

Hint: look at question b). Note that question b) refers to the speed of the truck at the highest point.

I think this is sufficiently close to homework that it should be moved to that section.

beamie564
PeroK said:
It is gravity and, as you conclude, at this point (for an instant) the truck is in free fall. To see what's happening, imagine the track was only the right half of the loop, so that after the highest point, there is no track to the left. What would happen to the truck?

Hint: look at question b). Note that question b) refers to the speed of the truck at the highest point.
I assume, the truck will go straight for a while because of inertia. So how exactly am I going to put this into words?
"It is gravity that keeps the truck in it's circular motion and inertia that drives it away from the circle. The net sum of these two forces is what keeps the truck in circular motion"

vinci said:
I assume, the truck will go straight for a while because of inertia. So how exactly am I going to put this into words?
"It is gravity that keeps the truck in it's circular motion and inertia that drives it away from the circle. The net sum of these two forces is what keeps the truck in circular motion"

Inertia is not a force. The truck won't go straight for a while, it will fall in a parabola. Try drawing parabolas for the truck:

a) If it's still moving fast at the top of the loop.

b) If it's moving "too slowly" at the top of the loop.

We're still imagining there is no track on the left here, so the truck is falling through the air.

Centripetal force is the net force. What two forces sum to make it? Try drawing a free body diagram to the car.

CWatters said:
Centripetal force is the net force. What two forces sum to make it? Try drawing a free body diagram to the car.

The problem here is that the question deals with a special situation at the top of the loop where there is no normal force: this is an assumption in the question. There is only gravity acting on the truck at this point. The conundrum is why the truck doesn't fall off the track at this point.

CWatters
Fair enough but I think it helps to understand the general principles first. It's useful to think of the centripetal force as the force required for an object to move in a circle and then consider what must happen to change the radius of motion (not to be confused with the radius of the track).

PeroK said:
There is only gravity acting on the truck at this point. The conundrum is why the truck doesn't fall off the track at this point.

At that point the truck is running parallel to the track. Its velocity is purely horizontal, as is the track(d(track)=0) ;there's no net force holding it there, nor one pushing it apart. It is neither rising, nor falling. Just like a tossed ball, at the apex, it's "weightless".

The definitions of both centripetal and centrifugal tend to be a little vague, but yes, gravity is providing a positive centripetal force on the upside down bit, and a negative centripetal force on the rightside up bits.

@PeroK If I imagine the two cases in both of them the horizontal motion will be unaffected by gravity (i am assuming there is no air resistance), the truck will vertically accelerate downwards at the same time until it hits the ground. In case a) where it is moving too fast the point x-intercept of the parabola will be far away from the track and in the latter case it will be closer to the tracks. I know that there is supposed to be a case where it will travel in a circle but i can't fathom how that would be the case and why it would not just get out of the orbit. We are not talking about geo-stationary satallites who are in orbit because of the curvature of Earth falling under their path. @hmmm27: So the answer to question 'a' would be gravity, Right?

hmmm27 said:
The definitions of both centripetal and centrifugal tend to be a little vague, but yes, gravity is providing a positive centripetal force on the upside down bit, and a negative centripetal force on the rightside up bits.

This isn't helping.
vinci said:
@PeroK If I imagine the two cases in both of them the horizontal motion will be unaffected by gravity (i am assuming there is no air resistance), the truck will vertically accelerate downwards at the same time until it hits the ground. In case a) where it is moving too fast the point x-intercept of the parabola will be far away from the track and in the latter case it will be closer to the tracks. I know that there is supposed to be a case where it will travel in a circle but i can't fathom how that would be the case and why it would not just get out of the orbit. We are not talking about geo-stationary satallites who are in orbit because of the curvature of Earth falling under their path.

You're over-thinking this a little. If, at the top of the loop, the truck has sufficient speed, then it will fly off in a parabola outside where the circular track would be. So, when you add the track, the truck is forced into the circular path.

If, however, the truck is moving too slowly, the parabola will lie inside the circle. In which case, when you add the track, the truck will leave the track until at some point it crashes back into it.

That's one explanantion why there is a minimum speed required at the highest point. If the truck isn't moving fast enough, then it does fall off the track.

vinci said:
So the answer to question 'a' would be gravity, Right?

Oh, I wouldn't say that if I was getting marked on it, since gravity at no time is actually responsible for keeping it in the curve. It's not the Moon being kept from flying off by the Earth's gravity.

As far as visiualization is concerned, the (half) parabola's pretty good. Say you have a raised horizontal track that joins to a quarter circle track which plunges into the ground. If you push the cart fast enough that it flies off the end without actually touching the quarter circle, then that's a speed you want to be at to make it through the top of the loop in the diagram.

vinci said:
a) Explain what provides the centripetal force to keep the truck moving in a circle.
vinci said:
So the answer to question 'a' would be gravity, Right?

The more I look at this, the less I like question a).

There is only one point on the track (at the top) where gravity is centripetal. Hence, gravity alone acts only instantaneously. There is no finite period of time where gravity alone acts, so you cannot really say that gravity causes anything to move in a circle (unless it's an orbit, which is a very different scenario). You can only describe motion as being in a circle over a finite period of time. All motion is instantaneously in a single direction.

My example of the missing track proves this point. The motion is the same at the top of the track whether the track is missing on the left or not. If it's missing, the truck follows a parabola; and if the track is present, it moves in a circle. So, you cannot say what shape it's moving in at the top.

Anyway, the important point is to understand the physics. It is possible, if you get the speed of the truck just right, for there to be no normal force (instantaneously) at the top. Any faster and there is a normal force; any slower and the truck will leave the track. That's the key point here.

The expected answer to a) is gravity, but the wording of the question is perhaps dubious.

I doubt the expected answer would be gravity, since gravity never actually helps keep the cart in the track.

hmmm27 said:
I doubt the expected answer would be gravity, since gravity never actually helps keep the cart in the track.

Then you'd be wrong! Gravity is a centripetal force at the single point at the top of the loop. In this case, it is the only force at that point.

Then you'd be wrong!
I said I doubted it would be the expected answer. [edit: I am allowed to change my mind, right ? yes, I imagine it would be the expected answer : I misread the problem]

PeroK said:
it is the only force at that point.
If you're going to define "centripetal force" as simply something that works against inertia, then a vector component of gravity applies to the entire upside-down bit. This could be demonstrated as an increasingly flimsy track that would break apart if the full "centrifugal force" wasn't compensated for.

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hmmm27 said:
I said I doubted it would be the expected answer.If you're going to define "centripetal force" as simply something that works against inertia, then gravity applies to the entire upside-down part of the track as a vertical vector component. This could be demonstrated as an increasingly flimsy track that would break apart if the full "centrifugal force" wasn't compensated for.

This is a homework thread, in which we are trying to help the OP.

If you want to ask questions about what is and is not a centripetal force, you need to start your own thread. Leave this one for the OP's questions.

right, sorry, got sidetracked - given the simplicity of the 'b' problem, I figured 'a' would be just a general question and didn't read it through properly; answer edited.

vinci said:
I know that there is supposed to be a case where it will travel in a circle but i can't fathom how that would be the case and why it would not just get out of the orbit. We are not talking about geo-stationary satallites who are in orbit because of the curvature of Earth falling under their path.

Okay, forget parabolas for a minute.

The truck sticks to the track because of the normal force; in space, the normal force would only be created by the truck running into the curvature and getting squished into the track with a centripetal acceleration ##a=v^2/r##. Give the slightest push to the truck and it will go 'round and 'round the track forever. But, it's not an orbit in the sense of gravitational pull, as you point out. (As an aside, don't dwell on it, the track could be elliptical, or it could be some massive 3D assembly of twisty inside curves)

On Earth, however we have gravity. It's the most common source of the "normal force", eg. between the person and the chair he or she's sitting on. If you bring the track down to Earth then gravity will affect whatever's put on it, as it does everything, including the normal force of truck on track.

In order to stick to the track, the normal force has to be > 0. The total force perpendicular to the tangent - rotational force plus gravitational component - must be positive. In terms of acceleration ##a_r + g_θ > 0## needs be achieved to keep it stuck to the track.

For example, at the midpoints of the circle halfway up/down, gravity - a purely vertical force - has no effect on the normal force - which is purely horizontal at that point - so the rotational velocity only has to be greater than 0. At the bottom of the circle (indeed any point in the bottom semicircle), gravity will always contribute to the normal force to an extent, the most at the bottommost point.

However at the top the full force of gravity is at play against the normal force, so ##a_r > g## is the only thing that will keep it on track.

(As an aside, I can't help but note that if the truck had a motor and brakes, it could stop at the midpoints, before continuing. Alternatively It could even loop around at a constant angular velocity. But it's easiest just to let it zoom up to the top, slowing down all the way, then come back down, speeding up all the way.)

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The truck is not "falling" [edit: in the usual sense of a downward velocity] : the slope of its motion is 0: no vertical component. Furthermore, it has not and will not escape the track: even though the normal force is 0 (ie: it's not stuck to the track), since it isn't falling it isn't leaving, and the instant after that instant (pardon the semantics). the vertical component of the rotational force is greater than g, and it continues to not fall.

Having addressed most of the OP's concerns (reposted to avoid up/down paging)

vinci said:
I am trying to study Circular Motion for my exams and I'm kind of unsure about one question. The question asks what's keeping the truck in circular motion. It has to be gravity I know, but gravity being directed towards the center, shouldn't that just result in the truck falling? What keeps it INTACT to the track? I am giong to quote the actual question now.

"Figure 18.17 shows part of the track of a roller-coaster ride in which a truck loops the loop. When the truck is at the position shown there is no reaction force between the wheels of the truck and the track. The diameter of the loop in the track is 8.0 m.

a) Explain what provides the centripetal force to keep the truck moving in a circle.
b) Given that the acceleration due to gravity g is 9.8 m s-2, calculate the speed of the truck.

"

we can see that, there is no explicitly stated connection between the diagram and problems a and b, concerning the position of the truck, however one can safely assume that b refers to the diagram.

a on the other hand is problematic, since the wording states that there is something "providing a centripetal force to keep the truck moving in a circle" and at that point there is not - gravity is present on the centripetal vector, but it's doing squat as far as moving in a circle is concerned.

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hmmm27 said:
The truck is not "falling"
Depends what you mean by falling. In everyday use, it would probably be taken to imply moving downwards in free fall, but in a physics context it would be more usual just to require it to be in free fall. Throw a stone up at an angle. It is in free fall all the time until it hits the ground.
hmmm27 said:
gravity is present on the centripetal vector, but it's doing squat as far as moving in a circle is concerned
Nonsense.
The question states that there is no normal force at this particular point in the track, and yet the truck retains its circular trajectory. There must be a centripetal force to achieve that, and gravity is the only source.

vinci and Merlin3189
Interestingly, if we focus solely on the question of the OP
vinci said:
what's keeping the truck in circular motion
then I think it is correct to say that gravity plays no role in keeping the truck in circular motion. With a track like this, the only thing forcing the truck to travel in a circle is the track. The only thing gravity can do to the locus of the truck is to disrupt the circular motion, by making the truck fall downwards off the track if it's not going fast enough.

The discussion seems to have moved from the original question to the related but different question of what is the source of the centripetal force at the top of the track. The answer to that question is 'only gravity', but that doesn't mean that without gravity the truck would not follow the circle. In fact, without gravity, the truck would be traveling faster at the top and so would have a greater normal force than ##9.8 ms^{-2}## exerted on it by the track.

In the absence of gravity and friction, the truck would make a complete loop and then continue, without ever varying speed.

In the absence of gravity but with friction present, the truck would continue around the loop, getting gradually slower, until it came to a stop because of friction. It would then simply remain wherever it had stopped. Whether it completed the loop before that happened would depend on the initial speed and the coefficient of friction.

In the presence of gravity and friction, the truck starts slowing as soon as it enters the loop. If it comes to a stop before it becomes vertical, it will just roll backwards down the way it came. If it gets past there but its speed ever falls below that needed to keep it on the track, gravity will make it fall off the track. In the absence of friction, that would have to happen - if it happens at all - before it reaches the apex. With friction present, it could happen after the apex.

Actually, as PeroK pointed out, the question doesn't actually make sense, because it is asking what is keeping the truck in circular motion at the single point that is the apex. The question is meaningless because in physics there is no such thing as an instantaneous effect, so the absence of either gravity or the track at such a dimensionless point would make no difference whatsoever to the locus of the truck.

vinci
andrewkirk said:
I think it is correct to say that gravity plays no role in keeping the truck in circular motion
As I wrote in post #22, the question is asking the student to explain how come it could remain on a circular trajectory even when there is no normal force. I.e., where is the centripetal force coming from? The answer is gravity. Sure, if you took away gravity it would still move in a circle, but then the normal force would be nonzero. The question is not asking whether gravity is necessary to produce the circular motion.
andrewkirk said:
it is asking what is keeping the truck in circular motion at the single point that is the apex... the absence of either gravity or the track at such a dimensionless point would make no difference whatsoever to the locus of the truck.
The curve is presumed smooth, with a continuous second derivative. Therefore it has a centripetal acceleration at all points, including the apex. Your argument is similar to Zeno's arrow paradox.

vinci and Merlin3189
Explain what provides the centripetal force to keep the truck moving in a circle.

Nothing. The force of gravity will 'keep the truck moving' in a parabola.

At the next instantaneous point over - where we can define the motion as being definitively circular - the normal force is reasserted.

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vinci
hmmm27 said:
Nothing. The force of gravity will 'keep the truck moving' in a parabola
It moves in a circle. It never ceases moving in a circle. Therefore, at all times, there is a centripetal force.
Even if an object moves in a parabola it has, at each instant, a centre of arc, so there is a centripetal force.
The centripetal force is the component normal to the velocity of the resultant of applied forces.
The only applied force at the apex is gravity. Ergo, gravity is providing the centripetal force.

haruspex said:
Even if an object moves in a parabola it has, at each instant, a centre of arc, so there is a centripetal force.

So you're contending that any force is "centripetal" in nature when it acts on an object ?

Otherwise, the only centripetal force in the problem is the normal force - gravity affects that, but is not a centripetal force in its own right.

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hmmm27 said:
So you're contending that any force is "centripetal" in nature when it acts on an object ?

The only centripetal force in the problem is the normal force. Gravity affects that, but is not a centripetal force in its own right.
A centripetal force is not an applied force, it is a resultant of applied forces. It is only the component normal to the velocity, so no, I am not saying any force is centripetal in nature.
Beyond that, I don't think there is universal agreement on its definition. Some would only use the term in circular motion, while others accept it in any motion where there is an instantaneous centre of arc. Others reject the term altogether, only referring to centripetal acceleration.

Certainly for a satellite in a circular orbit you would have to accept gravity as providing the centripetal force.

CWatters and Merlin3189
haruspex said:
Certainly for a satellite in a circular orbit you would have to accept gravity as providing the centripetal force.

Yes, and I accept that a point or line source force has an integral "centripetality"

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@haruspex I'm not sure whether we are in agreement or disagreement. My opinion is that the question is not a proper science question, and should be relegated to the philosophy department. Physics questions are about formulas, equations and measurements, not about causes. 'Cause' is a rubbery term that can mean almost anything. I think your comparison of the question to Zeno is apt, because the question is like Zeno's paradox statements - a confusion that arises from using imprecise words instead of precise equations. But unlike whoever wrote this question, Zeno had the excuse that calculus had not been invented yet.

When I am finally emperor of the world (why is it taking so long!) it will be forbidden for science lecturers to ask questions like 'what is keeping the truck moving in a circular path' as opposed to 'identify all objects that exert a force on the truck and find the magnitude of each such force'. The answer to the first question, as with all other such questions, is 'the Big Bang'.

vinci
So, at the end of the day, all the help we can give the OP is

When answering 'b', be sure to specify the domain, ie: "at the apex" or "within the circle".

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vinci
I think this all raises more questions than there already were. I am going to go with 'gravity' while answering this and hope they just don't bring this question in the exam.(Too many confusions, specifically because of this part 'When the truck is at the position shown there is no reaction force between the wheels of the truck and the track.')

vinci said:
(Too many confusions, specifically because of this part 'When the truck is at the position shown there is no reaction force between the wheels of the truck and the track.'
There's actually no problem with that statement. It is a clear, meaningful, scientific statement, because it talks about the forces that two bodies exert on one another. Further, it is correct.

The problem is question (a), which is unclear. A good exam strategy when faced with unclear questions is to just write down your understanding of the situation. For example here you could write something like:

Excluding friction and air resistance, the net force on the truck can be decomposed into components normal and parallel to the track.

The parallel component varies over time according to this formula <write formula> and causes the speed of the truck to vary. That component is equal to the component of the gravitational force on the truck that is parallel to the track. It is zero when the truck is at the point shown.

The normal component is always equal to <insert formula> and is a centripetal force as a result of which the truck moves in a circle. It is the sum of the normal force exerted by the track and the component of the gravitational force on the truck that is normal to the track. The former varies over time and is equal to <insert formula>. It is zero at the point shown, so the magnitude of the gravitational force is equal to that of the centripetal force.​

It would be hard for them to mark you down with an answer like that.

vinci said:
Too many confusions, specifically because of this part 'When the truck is at the position shown there is no reaction force between the wheels of the truck and the track.'

I don't get the confusion. It would be redundant to add "and only at that position" since, if at any other point there is "no reaction force" (due to the interaction with gravity), then the truck never reaches the apex.

At the apex the truck's motion is parallel to the track, at a distance of zero. Gravity's pull has canceled rotation's push. 'b' solves for the velocity necessary to achieve that.

Do feel free to worry about if it's actually "on" or "off" the track, though. Is the glass half empty or half full ? Is zero a positive or negative number ? Yes, we have no bananas.

Everybody else is annoyed at 'a', to which I propose...

Gravity provides a pseudocentripetal force to the truck at the apex. While the vector is equal to that of the track and the force is perpendicular to velocity, the motion due to gravity is elliptical at that velocity, and is not contributing to circular motion.

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## 1. What is circular motion?

Circular motion is a type of motion in which an object moves along a circular path. This means that the object's distance from a fixed point remains constant, but its direction constantly changes.

## 2. What is centripetal force?

Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle. It is responsible for keeping the object moving along the circular path.

## 3. What is the source of centripetal force in circular motion?

The source of centripetal force in circular motion can vary depending on the specific situation. In some cases, it may be due to gravity, such as in the case of a planet orbiting around the sun. In other cases, it may be due to tension in a string or friction between the object and the surface it is moving on.

## 4. How is centripetal force calculated?

The magnitude of centripetal force can be calculated using the formula F = mv^2/r, where F is the force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.

## 5. Can an object in circular motion have a constant speed?

Yes, an object in circular motion can have a constant speed as long as its direction is constantly changing. This is because speed is a measure of how fast an object is moving, while velocity takes into account both speed and direction.

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