# Uniform Circular Motion and Centripetal Acceleration (non-intuitive)

babaliaris
Never heard of tangential acceleration yet. I just know that a particle that is doing uniform circular motion has only one acceleration, the centripetal acceleration. This is what the book told me so far. Next chapter is kinetic energy.

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Never heard of tangential acceleration yet. I just now that a particle that is doing uniform circular motion has only one acceleration, the centripetal acceleration.
Very well, but in the situation you examining at post #30, the particle does non-uniform circular motion... that's the whole "catch"... So better to avoid this problem till you learn about tangential acceleration.

babaliaris
I see... I will wait then until i reach the appropriate chapter. Once again I'm rushing

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+1

It's not often you need to calculate the net acceleration when the motion is non-uniform. Try easier problems first.

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Again, math is your friend. If I understand the problem right, you have a circular loop in the ##x_1 x_2## plane. For convenience I put its center in the origin of the coordinate system. The gravitational acceleration is ##\vec{g}=-g \vec{e}_y##. Now the motion on the circular loop most simply is described with polar coordinates,
$$\vec{r}=r \cos \phi \vec{e}_1 + r \sin \phi \vec{e}_2.$$
The equation of motion is
$$m \ddot{\vec{r}}=m \vec{g} \, \Rightarrow \, \ddot{\vec{r}}=\vec{g}.$$
The only thing you need is the 2nd derivative wrt. time.

Then fortunately the force is conservative, i.e., it has a potential
$$V(\vec{r})=-m \vec{g} \cdot \vec{r}=m g x_2=m g r \sin \phi.$$
Now energy is conserved, i.e.,
$$E=\frac{m}{2} \dot{\vec{r}}^2+m g r \sin \phi=\text{const}.$$
With the chain rule (and because ##r=\text{const}##) you have
$$\dot{\vec{r}}=-r \dot{\phi} \sin \phi \vec{e}_1 + r \dot{\phi} \cos \phi \vec{e}_2$$
and
$$\dot{\vec{r}}^2=r^2 \dot{\phi}^2.$$
Now it should be very easy to solve your problem, using the energy conservation law above.

Friend of Kalina
1. The ball on the string doesn't fall to the center because it starts with a velocity tangent to the circle. The string pulls it into the center of the circle, the velocity imparts an inertial component to keep going straight, the result is that the ball turns along the circumference of the circle.

2. The ball doesn't fall to the ground because the string is holding it up. If you stand to the side, you will see that the string is not parallel to the ground, but is instead at an angle. The total force on the string is composed of the horizontal acceleration, keeping the ball going in a circle, and the vertical acceleration countering gravity. The flatter you keep the string, the higher the force required; centripetal force is linked by the angle of the string, but also increases, and the ball goes faster.

Oops. Somehow failed to notice I was still on page 1 of 3 and that question had been resolved. Sorry.

babaliaris
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I'm in the chapter of Uniform Circular Motion and I have a hard time understating centipetal acceleration. Until now I knew that acceleration describes "how fast velocity changes in magnitude" except projectile motion because when it reaches maximum height then g is going to change the direction on the Vy and make it move downwards but now in this kind of motion, acceleration describes only "how fast the direction of the velocity changes" and not the magnitude. Ok I understand that but what I really don't understand is when I'm in a car and take a turn I feel pushed in the opposite direction of the circles center. But acceleration is pointing at the center of the circle
You probably have the answer but...
When a car accelerates forward, you feel like you are being pushed, back into the seat.
When a car brakes heavily in an emergency stop (it is accelerating back) you feel you are being thrown forward,
Why should it feel unusual that when a truning car is accelerating towards the centre of the circle (centripetal acceleration) you feel like you are being thrown outwards, away from the centre.
The real truth is that you are not being pushed or thrown in any of those directions - it just feels like you are. We are often deceived. We can even feel (momentarily) lighter when an elevator starts to move, then (momentarily) heavier when it stops again.

babaliaris, CWatters and Delta2
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You probably have the answer but...
When a car accelerates forward, you feel like you are being pushed, back into the seat.
When a car brakes heavily in an emergency stop (it is accelerating back) you feel you are being thrown forward,
Why should it feel unusual that when a truning car is accelerating towards the centre of the circle (centripetal acceleration) you feel like you are being thrown outwards, away from the centre.
The real truth is that you are not being pushed or thrown in any of those directions - it just feels like you are. We are often deceived. We can even feel (momentarily) lighter when an elevator starts to move, then (momentarily) heavier when it stops again.
No! What you feel in the car is that you get pushed forward by the force imposed on you by the seat!

It's the same thing in a merry-go-around: You feel the push of the force imposed on you by the seat towards the center of the circle, i.e., the centripetal force.

No! What you feel in the car...
"Feel" can mean different things:
- Direct sensation of contact forces (what you probably mean)
- Interpretation of your movement relative to the car (what Peter probably means)

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Maybe a physicist "feels" differently from being trained to think differently. ;-)).

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In a carnival scrambler ride, there's a lot of centripetal acceleration, but 2 people sit side by side in the ride with no seat belts. If the riders end up pushing against each other, the outside rider exerts a centripetal force on the inside rider, and the inside rider exerts a centrifugal reaction force to the outside rider. This is one example where a centrifugal reaction force can be felt.

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No! What you feel in the car is that you get pushed forward by the force imposed on you by the seat!

It's the same thing in a merry-go-around: You feel the push of the force imposed on you by the seat towards the center of the circle, i.e., the centripetal force.

I am quite aware of what forces are actually acting on you - I was using "feel" as the sensation you experience. Indeed if you watch an unbelted child sitting in the back seat of the car they do not feel a centripetal force from the seat - they eventually feel the centripetal force from the door when the have finally tumbled that far. They don't know to hang on - especially if they have been brought up to always wear a seat belt and have just omitted this time. The seat belt can supply the centripetal force need without the "outward tumble" - which is, I know, just continuing (approximately) in a straight line.
By the way, another way to make the seat push forward on you is to have someone else push you into the seat - while the car is stationary. It that case you still don't notice the seat pushing, but wonder why the person is pushing you. When the car accelerates, you still don't notice the seat pushing, you just wonder who is pushing you into the seat this time. One reason to study Physics is so you can explain that there is nothing pushing you into the seat, it is just a ficticious force that has appeared in the non-inertial frame you have briefly found yourself in.

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there is nothing pushing you into the seat, it is just a fictitious force that has appeared in the non-inertial frame you have briefly found yourself in.
When a car is accelerating, you have a Newton 3rd law pair of forces, the seat exerts a forwards force on the person, the person exerts a reaction force on the seat. The reaction force is a real force in any frame of reference.

Using the car as an accelerating frame of reference, the fictitious force is the force that would appear to accelerate a body in "free fall" with respect to the accelerating frame of reference, and would not be part of a Newton 3rd law pair of forces. The Newton 3rd law pair of forces are still the same, seat exerts forward force on person, person exerts backwards force on seat.

vanhees71 and Delta2
...person exerts backwards force on seat.
Yeah, we must not forget how the seat feels about it.

CWatters
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When a car is accelerating, you have a Newton 3rd law pair of forces, the seat exerts a forwards force on the person, the person exerts a reaction force on the seat. The reaction force is a real force in any frame of reference.

Using the car as an accelerating frame of reference, the fictitious force is the force that would appear to accelerate a body in "free fall" with respect to the accelerating frame of reference, and would not be part of a Newton 3rd law pair of forces. The Newton 3rd law pair of forces are still the same, seat exerts forward force on person, person exerts backwards force on seat.
Nothing wrong with those Newtons 3rd law Forces, other than why they are happening.
I contend a person only expects to apply a force on the seat, if they decide to push, or can simply explain why they are doing it.
In a stationary car, If a third person pushes an occupant, the occupant is not surprised that they therefore apply/transfer a push to the seat, and they will feel the seat pushing back. That person does NOT feel themselves pushing back on the third person - only the third person feels that force.
When the occupant is in the accelerating car, they feel the seat pushing forward, but may be at a loss to explain why the seat is doing that - because in that situation there is no third person instigating the action/reaction. Unfortunately, after just a few seconds, the car ceases to accelerate and the occupant quickly loses interest in the situation except to think "wow, something really pushed me into that seat just then", and perhaps include "it must have been the engine". Remember, we are not talking about a physics person explaining what is happening, we are talking about an "innocent" wondering what just happened.

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When the occupant is in the accelerating car, they feel the seat pushing forward, but may be at a loss to explain why the seat is doing that
A person undergoing acceleration is going to feel real forces internal to their body and feel compression forces at the points of contact with the seat. This is more clear in the case of greater acceleration such as a graviton / rotor spinning carnival ride that pins the persons (victims) against the wall while the floor drops, or in a vehicle capable of pulling more than 1 g.

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A person undergoing acceleration is going to feel real forces internal to their body and feel compression forces at the points of contact with the seat. This is more clear in the case of greater acceleration such as a graviton / rotor spinning carnival ride that pins the persons (victims) against the wall while the floor drops, or in a vehicle capable of pulling more than 1 g.
Yes they will feel the force, but if they are a non-physics person (as are our students when they begin to stud the subject) they won't necessarily understand why that force suddenly came about. Seats and walls don't usually start pushing you unless you start pushing them - in these cases, the seat/wall seems to instigate the entire action/re-action.

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Yes they will feel the force, but if they are a non-physics person (as are our students when they begin to stud the subject) they won't necessarily understand why that force suddenly came about. Seats and walls don't usually start pushing you unless you start pushing them - in these cases, the seat/wall seems to instigate the entire action/re-action.
I think most non-physics persons get a sense of these forces when riding roller coasters that include high g (accelerating) turns.