# Why is centrifugal 'force' perpendicular to line of inertia

I know that centrifugal is labeled a fictitious force only arising in a rotating reference frame, but I still struggle to understand the forces at play intuitively in tethered rotating bodies.

I've always been told that the reason a slingshot works is because the ball wants to go in a straight line. "Yeah cool", I've always just nodded my head but never been able to understand it really. Intuitively why wouldn't that straight line be in the direction of acceleration - which would be the tangent line at the arc segment at release point. The direction of acceleration is the same direction for the propagation of inertia usually!? The fact that its perpendicular really makes it look like its own force! It appears as if inertia in one direction is instantly translated to its perpendicular!

Also, (this might be two questions), if every force has an equal and opposite one, then why is centripetal force considered a legitimate force but its opposite (centrifugal) a fictitious force? Centripetal ALSO only occurs in a rotating reference frame right?

Everything I've read about these two "forces" over the years is always included the same old catch phrases, but for some reason I can't understand it intuitively. I've seen the pictures of the boy with the ball on the string and the arrows pointing inwards for centripetal and outwards for centrifugal but to me thats just labelling an observed effect that is occurring. It doesn't explain WHYYYYY :). I've parroted off the mechanics to people many times with zero intuitive understanding as to why its all happening.

Is the explanation related to inertial properties?

• – Qmechanic Mar 24 '14 at 23:15
• "why wouldn't that straight line be in the direction of acceleration..." What's wrong it is in the direction of acceleration. e.g pad (in which the object to be thrown is held) moves with a speed greater than that of the object. When the pad's speed lowers than that of the object the object starts to move in a straight line. compare the situation with a fast bowler of cricket about to ball after taking a long run up. The ball moves in a curved path untill the hand of the bowler achieves a speed less than that of the ball itself. – user31782 Mar 27 '14 at 5:01
• "The direction of acceleration is the same direction for the propagation of inertia usually!?" Not usually, It is only the case when the appplied force is in the same direction as that of the propogation of inertia. Haven't you read Newton's laws? The direction of acceleration is always the same as that of the net Force acting on an object. – user31782 Mar 27 '14 at 5:04
• This question bugged for almost 2 months. No teacher at my school was able to see what I was asking. After a few diagrams by myself, I was able to get it rightly. (Its about an year since...) @Neuneck provides the solution which I had thought. It, IMHO, is the right answer. – user3459110 Mar 27 '14 at 13:08
• – TanMath Oct 28 '15 at 18:26

I think your confusion with the slingshot is this: when you move your hand in circles to keep the slingshot moving in a circle you need some force, so you feel the weight of the projectile. But you where taught that according to Newton, things want to keep going in a straight line and inertia is this tendency to "not want to move or change direction". So since the slingshot is always changing the direction of motion in direction of the tangent line of the circle, you should feel some inertia in this direction. This is why you asked:

"Why does there seem to be zero inertia along the tangent line when that is the direction it is moving at any moment?"

Well you're still correct when you say that you need a force to change the direction of motion and that inertia is the tendency to withstand that change. But the important thing here that in order to have the projectile move in a circle you have to exert a force on it in the direction of the center of the circle. When you're doing this, you change the direction of motion (i.e. the momentum) in such a way that the projectile moves in a circle and the direction points along the tangent of the circle. But since the force is pointing inward, you also feel the inertia in this direction.

When you let go of the projectile, the force (which changes the direction of motion) is gone and it moves along the tangent line. (Just like the others have explained it to you)

Also, if every force has an equal and opposite one, then why is centripetal force considered a legitimate force but its opposite (centrifugal) a fictitious force?

I have a very good answer. This our professor told us and the moment I heard it the distinction between fictitious and real forces was clear to me:

Real forces are resulting from the interaction of two bodies. With fictitious forces, this is not the case.

This is why you need a centripetal force to keep the ball moving in circles: you're interacting with a second body - so you feel his inertia. But if you had a hollow ball and a tiny person living in it, this person would feel a centrifugal force. The tiny person is in no way interacting with you, it's just in his reference frame that this force shows up.

Centripetal ALSO only occurs in a rotating reference frame right?

Yes, that's true. But like I said, it's a force to keep another body moving in a circle.

• The centripetal force is a real force and exists in any reference frame. The equal and opposite forces refers to ration forces eg. you hand pull on the rope with 5 lbs towards your hand, and the rope pulls on your hand with 5 lbs away from your hand; the rope pulls on the rock with 5 lbs towards your hand and the rock pulls with 5 lbs of force away from your hand. Notice here that the centripetal force on the rock was not paired with the centrifugal force. – Rick Jan 30 '15 at 20:37

why wouldn't that straight line be in the direction of acceleration

Why do you think the acceleration line be in the direction of tangent? The tangent is where a body would have kept moving if the rope didn't pull it. so the vector of speed changes towards... Where the rope is attached, i.e. perpendicularly. Acceleration is the change in velocity vector. Intuitively (habitually) people think of acceleration in terms of the same direction but different speed, but in fact acceleration is any change in velocity, whether it's the speed or the direction.

• Yes Sir.. Accelration is change in velocity (magnitude or direction)! – mcodesmart Mar 7 '14 at 6:15
• @aksakai I get that you are assuming that the obvious is in deed obvious. I know that the centrifugal 'force' pulling the rope taut is manifested because its getting a constant course correction from the curve of the orbit. The tangent line is the closest extrapolation of a straight line from the curve so thats where you'd expect the inertia to propagate? Why is it then 90 degrees? Why does there seem to be zero inertia along the tangent line when that is the direction it is moving at any moment? – Mike S Mar 7 '14 at 7:14
• 90 degrees thing can't be explained in words, really. you have draw, consider infinitesimal changes etc. the only blah-blah argument to be done is that the rope is really the only thing that pulls the body, it's pulling to the center. it can't pull forward or sideways. – Aksakal almost surely binary Mar 7 '14 at 14:03
• @MikeS " I know that the centrifugal 'force' pulling the rope taut is manifested because "- Do you mean centripetal force? The centrifugal force is not manifested because of this reason. – user31782 Mar 27 '14 at 5:13
• At any given time a body is moving along the tangent, where the inertia is heading too, but the force is towards the center – Aksakal almost surely binary Mar 27 '14 at 11:16

Fictitious forces naturally arise in non-inertial (accelerating) reference frames and you have to be careful with them. In this example, it only leads to confusion.

$F = ma$ tells us that when the (total) force is zero, the object will continue in a straight line. It's not the centripetal force, but the absence of a centripetal force that makes the object fly away.

Centripetal force does come with an 'equal and opposite force' - in your example, you have to exert a force to keep your end of the rope in the centre of the rotation.

The direction of acceleration is the same direction for the propagation of inertia usually!

This is where you are wrong. Direction of propagation is the direction in which the body is currently moving or rather, changing position. So it is the direction of infinitesimal displacement $d\vec x$ at that instant. Now what makes you think that direction of acceleration or $\vec a$ is that same direction?

By definition, $$\vec v = \frac {d\vec x}{dt}$$ So the velocity or $\vec v$ should have the same direction as $d\vec x$, not $\vec a$.

Intuitively, you can say that when the string is pulling the ball towards the center, it is having a direct effect on the ball's velocity, not the ball's position. So that centripetal force is changing the direction of the ball's velocity to maintain its direction tangent to the circular path. But that force has no (direct) affect on the ball's position.

Only the velocity of the ball can affect its position. The ball always moves in the current direction of its velocity because its change in position is governed by the velocity(aka rate of change in position).

Another disproving example is projectile motion. When you throw a ball at an angle to the horizontal, the only acceleration it has is downward(gravity). So then according to you, the ball should only move downward and not forward. But it does move forward because it has a velocity in that direction, and velocity is what changes the position not acceleration. Acceleration changes the velocity, which in turn has an effect on position, but there is no direct effect.

Okay, let's lay some ground rules:

1. An object that does not experience any force will fly in a straight line
2. A force applied to an object will change its momentum toward the direction of the force.

Now, the trick with circular motion is that both the direction of motion and the direction of the force change simultaneousely, such that the inward cetripetal force will always be perpendicular to the motion of your ball. You see, the force is always inward. All it does is change the direction in which your ball is moving.

Btw. the equal and opposite force acts on the person holding the sling, that's why hammer throwers are such beefy guys (or gals :-P )

The acceleration is gone the instant you let go and the ball does what comes naturally and flies off in a straight line tangential to the circle it moved on before. The direction of the accelration is not important anymore, the ball could just as well have come from a cannon pointing that way. That's because the ball will just fly straight on, as long as no force acts. What was before doesn't matter!

Momentum is a vector, and the job of a force is to change it, in magnitude and direction. The ball has tangential momentum at any point, and when the force is perpendicular only the direction of momentum is affected.

You can think of the force as a reaction force needed to keep the ball in a circle, and since it is perpendicular to the direction of motion, it does not add or remove power to the system. To get a slingshot going, you need to move your hand back and forth, in order to make the tension not perpendicular to the motion and add power to the ball. Power as a scalar is the dot product between force and velocity.

Once the tension is released, the ball will follow the direction of its momentum, since there are no forces acting (ignore gravity). Newtons 2nd law is all you need to understand this problem.

There are several issues occurring here- none of which require a "fictitious force" to resolve.

1) the acceleration on the shot cannot operate in the direction of it's velocity as the shot in the sling maintains a constant distance from the slinger while it is being "slung".

2) there is no "centrifugal" force on the shot... the shot attempts to follow newton's laws of motion and depart in a straight line upon it's velocity vector... however this is restricted by the tension in slingshot itself which is in turn passed on to the slinger.

3) the proper force balance equation (action equal and opposite) for the shot includes the forces both on the shot and the slinger (careful examination of real slingshot videos show the center of balance of the slinger and the motion of the shot move together- equally and oppositely in harmonic motion)

4) upon release of the shot the forces are zero (except for gravity) and the shot flies off on a trajectory given solely by the velocity at this time. Note that for an object constrained to move in a circular motion the velocity vector is tangent to the circle at all times.

A particle does not have to move in the direction of the acceleration. Acceleration is change in velocity, so the change in velocity is in the direction of acceleration. Velocity, being a vector, obeys certain Laws of vector addition(see traingle law and parallelogram law of vector addition). For example, if two forces(another vector) of equal magnitude are applied on a body at an angle of 90 degrees the resultant force will be at an angle of 45 degrees to both the original forces. Every action has an equal and opposite reaction, but not on the same body, otherwise the body wouldnt move. If you are twirling an object in a circle using a string, one force provides the centripetal acceleration for the object to move in a circle. The reaction force is the object pulling on the string- the tension in the string. If you have too heavy an object and you rotate it really fast the string will break. Consider youre in a car, moving along a circle. Your dashboard is frictionless. You have an object on the dashboard. As you move in a circle, due to lack of friction, the dashboard object moves in a straight line. To you, in the car, it appears as if a force is pushing the object away from the centre of the circle. There is no such force, but because your inertial frame is an accelerating one and you and the object move in different directions you have to make up a pseudo force to explain why the object moves and to fit with Newton"s laws of motion. I saw an animation about the car thing once, I"ll try to find it. The object flies out of the window!

The centripetal force can actually be measured. If you take the slingshot as an example, while rotating the end-mass you can measure a tension in the stings of the slingshot.

If you stop the motion of the end-mass at a certain point in time, you can observe a velocity of the mass, that is tangential to the circular path it is taking over time. The strings of the slingshot enforce a constant distance between the end-mass and the centre of rotation.

Newton's 2nd law of motion states that a body will remain at rest if the body was at rest and no force is acted upon the body. Or the body will remain in constant translational motion if the body was in motion and no force acts on the body.

So, when we apply Newton's 2nd to the frozen slingshot, if we remove the centripetal force -- we let the strings of the slingshot slip -- then the end-mass experiences no forces (neglecting gravity) and it will continue its motion. Thus, it will move in the direction tangential to the circle it was describing previously.

The direction of the inertia propagation is the direction of the velocity. Any acceleration can be applied to change the velocity. If the acceleration has the same direction of the velocity you will change only its modulo, but if you want to change the direction of the velocity (which is your "line of inertia") you need "a force that push from the side" or, as a physicist would say: "a perpendicular component of the acceleration". This component is called centripetal (it becomes a centrifugal apparent force in the a non-inertial frame of reference). We say that a force is centripetal (or centrifugal) if it does not affect the modulo of the velocity, but only its direction, such force cannot have a longitudinal component (which would change the modulo of the velocity) so it has to be always perpendicular to the velocity. That's it.

When OP said, ""Why does there seem to be zero inertia along the tangent line when that is the direction it is moving at any moment?"" The inertia is always away from you, so when you spin the sling and let it go when it is in front of you, that is the real trajectory of the shot, the circle path only exist while you are swinging it. The acceleration force has to appose that inertia while keeping it on a circle path; that is why you feel weight when you spin the sling.

When OP said, "Also, if every force has an equal and opposite one, then why is centripetal force considered a legitimate force but its opposite (centrifugal) a fictitious force?" The force the person is exerting while spinning the sling is the counter force to the inertia created by the centrifugal rotation. Centripetal force and torque are more like opposites to me.

Your question if I understand it is, " I've seen the pictures of the boy with the ball on the string and the arrows pointing inwards for centripetal and outwards for centrifugal but to me thats just labelling an observed effect that is occurring. It doesn't explain WHYYYYY :)"

Let me start with an equation. This is the equation for determining force (F=ma) Force = mass x acceleration

acceleration is the "force" that is felt in centrifugal force. (Gravity is expressed the same way).

The mass on the string has no "weight" (unless you are in gravity already) until a force of acceleration acts on it. When you start spinning the ball on the string and it is at velocity, the circular path creates constant acceleration in a new direction called centrifugal force. This acceleration is giving the mass weight. If you spin it faster and increase the force on the mass, the energy will transfer to the mass and give it more weight.

Because weight is just mass x gravity (weight being the same as Force in F=ma), centrifugal acceleration (the "a" in F=ma) replaces "gravity" in the equation to give mass weight.

The angle of deviation in centrifugal force is an acceleration in a new direction in relation to a point in space, which gives a net acceleration on the mass, meaning that it was trying to accelerate against you but you are accelerating it a different direction, requiring force (energy) and thus giving it its weight. Weight is the energy translated from terms of mass and acceleration, basically.

All forces you can feel are forces transferring energy, so the energy in your hand transfers to the mass at the end of the string and accelerates the mass towards a new direction and gives the mass weight.

• If you want to down vote, would you point out the error? – d w Oct 27 '15 at 18:18
• Incoherent and doesn't even try to address the question of directions that the OP asked. – dmckee --- ex-moderator kitten Oct 27 '15 at 18:37
• Am I reading his question wrong or something? OP said, " I've seen the pictures of the boy with the ball on the string and the arrows pointing inwards for centripetal and outwards for centrifugal but to me thats just labelling an observed effect that is occurring. It doesn't explain WHYYYYY :)" I thought this was the main question. – d w Oct 27 '15 at 18:47
• @dmckee I have attempted to clarify my answer. – d w Oct 28 '15 at 14:23
• @dmckee Is my answer still bad? – d w Oct 28 '15 at 18:58