# Centripetal acceleration vector

Purely mathematically, how does the expression for the centripetal acceleration a=v^2/R give information about the direction of the acceleration vector? I don't seem to notice any indication that the acceleration vector should be perpendicular to the velocity, or pointing towards the center.

It's more of a conceptual question, really: I understand that we get the result with the ASSUMPTION that the acceleration has such direction, but I am only wondering whether it should be explicit in the result.

Thank you.

jhae2.718
Gold Member
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The acceleration vector comes from performing vector kinematics. Let's consider a case of motion of a point mass in the plane.

Consider a rotating reference frame $b^+$ relative to an inertial frame $n^+$, where the ##\hat{\vec{b}}_1## unit vector always points toward the point mass. We can write a position vector of a point mass in the ##b^+## frame as:$$\vec{p}=r\hat{\vec{b}}_1$$where the ##\hat{\vec{b}}_i\text{'s}## are the unit vectors defining ##b^+##. In the plane, we can write the angular velocity of the frame ##b^+## relative to the ##n^+## frame as $\vec{\omega}_{b/n} = \dot{\theta}\hat{\vec{b}}_3$.

There is a neat result that we call the kinematic transport theorem. Simply put, it says that the derivative of a vector in one frame ##b^+## as viewed by an observer in another frame, say ##a^+##, is made up of two parts: 1) the change of the vector in the ##b^+## frame as viewed by an observer in the ##b^+## frame, and 2) the angular velocity of ##b^+## relative to ##a^+##. Mathematically,$$\frac{{}^a\text{d}\vec{r}}{\text{d}{t}} = \frac{{}^b\text{d}\vec{r}}{\text{d}{t}} + \vec{\omega}_{b/a} \times \vec{r}$$
We can apply this to our position vector to get an inertial velocity vector:\begin{align*} \frac{{}^n\text{d}\vec{p}}{\text{d}{t}} &= \frac{{}^b\text{d}\vec{p}}{\text{d}{t}} + \vec{\omega}_{b/n}\times \vec{p}\\ &= \dot{r}\hat{\vec{b}}_1 + \dot{\theta}\hat{\vec{b}}_3\times r\hat{\vec{b}}_1\\ \vec{v} = \frac{{}^n\text{d}\vec{p}}{\text{d}{t}} &= \dot{r}\hat{\vec{b}}_1 + r\dot{\theta}\hat{\vec{b}}_2\end{align*}
Apply the KTT once more to obtain the inertial acceleration:\begin{align*} \vec{a} = \frac{{}^n\text{d}\vec{v}}{\text{d}{t}} &= \frac{{}^b\text{d}\vec{v}}{\text{d}{t}} + \vec{\omega}_{b/n}\times \vec{v}\\ &= \ddot{r}\hat{\vec{b}}_1 + \left(\dot{r}\dot{\theta} + r\ddot{\theta}\right)\hat{\vec{b}}_2 + \dot{\theta}\hat{\vec{b}}_3 \times \left(\dot{r}\hat{\vec{b}}_1 + r\dot{\theta}\hat{\vec{b}}_2\right)\\ &= \ddot{r}\hat{\vec{b}}_1 + \left(\dot{r}\dot{\theta} + r\ddot{\theta}\right)\hat{\vec{b}}_2 + \dot{r}\dot{\theta}\hat{\vec{b}}_2 - r\dot{\theta}^2\hat{\vec{b}}_1 \\ \vec{a} &= \left(\ddot{r} - r\dot{\theta}^2\right)\hat{\vec{b}}_1 + \left(2\dot{r}\dot{\theta} + r\ddot{\theta}\right)\hat{\vec{b}}_2 \end{align*}
From this last result, we have the centripetal acceleration given by ##-r\dot{\theta}^2\hat{\vec{b}}_1## (this result may look more familiar if we define ##\omega \equiv \dot{\theta}## and write ##-r\dot{\theta}^2\hat{\vec{b}}_1 = - r\omega^2\hat{\vec{b}}_1 = -v^2/r\hat{\vec{b}}_1##). But we know that ##\hat{\vec{b}}_1## points outwards, so the centripetal acceleration must point toward the center.

You can easily extend this result to 3-D.

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• 2 people
If you differentiate the velocity vector of a particle undergoing uniform circular motion you should get a vector perpendicular to it pointing towards the centre.

A.T.
Purely mathematically, how does the expression for the centripetal acceleration a=v^2/R give information about the direction of the acceleration vector
You mean the net acceleration vector in uniform circular motion? If it wasn't perpendicular to velocity, the speed would change, so it would not be uniform circular motion. In this later case the velocity-perpendicular component of an arbitrary acceleration is often called "centripetal" (the other component being "tangential"). That makes "centripetal acceleration" perpendicular to velocity per definition.

Andrew Mason
Homework Helper
Purely mathematically, how does the expression for the centripetal acceleration a=v^2/R give information about the direction of the acceleration vector? I don't seem to notice any indication that the acceleration vector should be perpendicular to the velocity, or pointing towards the center.

It's more of a conceptual question, really: I understand that we get the result with the ASSUMPTION that the acceleration has such direction, but I am only wondering whether it should be explicit in the result.

Thank you.
Somewhere between Newton and the 20th century the convention developed to call the component of acceleration that is perpendicular to the velocity of the body "centripetal". I am not sure why that occurred. Newton appears to have coined the term in his 1684 treatise "De motu corporum in gyrum" (On the motion of orbiting bodies). It is a confusing term if it is applied to a body in something other than circular orbit.

AM

jhae, that was the precise answer I was looking for.

Andrew Mason, thank you, but I don't think the term is confusing at all. Mine was a question regarding the explicitness of the vectorial nature of the result.

Philip Wood
Gold Member
Think this is quick and straightforward...

Let $\widehat{\textbf{n}}$ be the unit vector normal to the plane containing the circle.
Then the velocity vector will be at right angles both to $\widehat{\textbf{n}}$ and to the radius vector, so (for anticlockwise rotation in a circle drawn on the page, with $\widehat{\textbf{n}}$ pointing out of the page)

$$\widehat{\textbf{v}} = \widehat{\textbf{n}} \times \widehat{\textbf{r}}$$
that is $\frac{\textbf{v}}{v} = \widehat{\textbf{n}} \times \frac{\textbf{r}}{r}$, that is $\textbf{v} = \frac{v}{r}\widehat{\textbf{n}} \times \textbf{r}$.

Remembering that v, r and $\widehat{\textbf{n}}$ are all constants:

$\frac{d\textbf{v}}{dt} = \frac{v}{r} \widehat{\textbf{n}} \times \frac{d\textbf{r}}{dt}$ that is $\frac{d\textbf{v}}{dt} = \frac{v}{r} \widehat{\textbf{n}} \times \textbf{v}$ that is $\frac{d\textbf{v}}{dt} = \frac{v^2}{r} \widehat{\textbf{n}} \times \widehat{\textbf{v}}$ so $\frac{d\textbf{v}}{dt} = \frac{v^2}{r} (-\widehat{\textbf{r}})$.

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Andrew Mason
Homework Helper
jhae, that was the precise answer I was looking for.

Andrew Mason, thank you, but I don't think the term is confusing at all. Mine was a question regarding the explicitness of the vectorial nature of the result.
My point is that the direction of the centripetal acceleration vector, $\frac{v^2}{r}\hat{n} \text{ where } \hat{n} \text{ is the unit vector normal to the velocity }$ is not necessarily the direction of acceleration. It is only the direction of acceleration for uniform circular motion. It is confusing because the term "centripetal" means "centre seeking" and $\hat{n}$ is not toward the centre, except for uniform circular motion.

AM

A.T.
$\hat{n}$ is not toward the centre
It is towards the instantaneous centre of path curvature.

Andrew Mason
Homework Helper
It is towards the instantaneous centre of path curvature.
I am not sure what that means. How is that point determined? For example, how would you determine the centripetal acceleration of a projectile in flight? If the point keeps changing it is confusing to call it centripetal. That's all I am saying.

AM

the expression for the centripetal acceleration a=v^2/R give information about the direction of the acceleration vector? I don't seem to notice any indication that the acceleration vector should be perpendicular to the velocity, or pointing towards the center.

If the centripetal acceleration is a=v^2/R you are dealing with a Circular orbit.

For a non-Circular orbit: $$a=\frac{|v|^2}{\rho} * u_n$$where $$\rho = \frac{1}{\kappa}$$ and ##\kappa## is the curvature. and ##u_n## is the normal.

Andrew Mason
Homework Helper
I understand how to determine the radius of curvature. The problem is that the radius keeps changing if the motion is not circular.

The issue is very simple: an object prescribing non-circular curved motion about a central point (zero torque) is always accelerating always toward that central point. It is incorrect to speak about acceleration toward an arbitrary point that it is not accelerating toward (let alone one that keeps changing). If the body is not experiencing a change in tangential speed, then the motion will be circular and the acceleration is indeed centripetal acceleration in the direction of the radius of curvature (which is constant). But if it is experiencing a change in tangential speed, the body's acceleration is not along the radius of path curvature. Its acceleration is always toward the central point.

For non-circular curved motion, it would be correct to refer to the component of acceleration in the direction of the radius of curvature, which is necessarily the component of acceleration normal to the velocity of the body. Since the body does not accelerate in the direction of the radius of curvature it is a wrong to call this vector the "centripetal acceleration".

One could call it an instantaneous centripetal component of acceleration . But since Newton called the acceleration toward the central point "centripetal" it is somewhat confusing to call it "centripetal" at all.

AM

Philip Wood
Gold Member
An interesting case is the acceleration of a planet at the extremes of the major axis of its ellipse. The centres of curvature at these extremes lie on the major axis, as done the Sun, but all in different places on the major axis. So the forces on the planet at either extreme are directed, as we know, towards the Sun, but co-incidentally also towards the centres of curvature! Therefore, I argue, we are allowed to equate the forces to $\frac{mv^2}{\rho}$. But, appealing to the symmetry of an ellipse, $\rho$ is the same at each end, A and P, of the major axis, so
$$\frac{F_P}{F_A} = \frac{v_P^2}{v_A^2}.$$
But it's easy to show from Kepler's equal area law that
$$r_Pv_P = r_Av_A.$$
in which $r_P, r_A$ are distances of the planet from the Sun at P and A.
So $\frac{F_P}{F_A} = \frac{r_A^2}{r_P^2}.$
which is the inverse square law!

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Andrew Mason
Homework Helper
An interesting case is the acceleration of a planet at the extremes of the major axis of its ellipse. The centres of curvature at these extremes lie on the major axis, as done the Sun, but all in different places on the major axis. So the forces on the planet at either extreme are directed, as we know, towards the Sun, but co-incidentally also towards the centres of curvature!
Of course. But at that point there is no tangential acceleration so the acceleration is indeed all toward the central point (which is the focus of the ellipse at which the sun is located), which is also the centre of curvature, and therefore, the acceleration toward the centre of curvature is indeed centripetal.

AM

Philip Wood
Gold Member
[...] the acceleration is indeed all toward the central point (which is the focus of the ellipse at which the sun is located), which is also the centre of curvature [...]
AM
Neither the centre of curvature at P nor the c of c at A is at either focus.

D H
Staff Emeritus
Therefore, I argue, we are allowed to equate the forces to $\frac{mv^2}{\rho}$.
No. You've cherry-picked the two points where this happens to work. What you can say is that ##\frac{v^2}{\rho}## is equal to the normal component of acceleration. But that's true for any space curve.

So $\frac{F_P}{F_A} = \frac{r_A^2}{r_P^2}.$
which is the inverse square law!
You already implicitly assumed an inverse square law force, so it's not that surprising that you derived an inverse square law force.

Philip Wood
Gold Member
DH Don't follow your criticism. (1) Of course I've cherry-picked these two points; they are the only two points relevant to my argument! (2) Where have I implicitly assumed the inverse square law? What I have assumed is Kepler's first Law (elliptical orbit with Sun at one focus) and Kepler's second law (equal areas in equal times). By cherry-picking two special points on the ellipse, I've deduced the inverse square law.

I've also assumed that the force on the planet is central, directed towards one focus, and is a simple power law. So my argument is in no way a substitute for Newton's proof that an elliptical orbit with equal areas in equal times implies an inverse square law force directed towards a focus.

Postscript: Post 14 now contains diagram in attachment.

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D H
Staff Emeritus
DH Don't follow your criticism. (1) Of course I've cherry-picked these two points; they are the only two points relevant to my argument!
Exactly. Your argument fails at any other points. That's cherry-picking.

(2) Where have I implicitly assumed the inverse square law? What I have assumed is Kepler's first Law (elliptical orbit with Sun at one focus) and Kepler's second law (equal areas in equal times). By cherry-picking two special points on the ellipse, I've deduced the inverse square law. I've also assumed that the force on the planet is central, directed towards one focus, and is a simple power law.
No, you haven't deduced the inverse square law. First off, you used the only two points where ##v^2/\rho = a## and used these cherry-picked points to make a universal deduction. That's not valid.

Secondly, you've deduced something that you already assumed. There are only two classes of radial central forces that can result in elliptical motion, force proportional to distance and force inversely proportional to distance squared. You've put the object responsible for that central force at one of the foci of the ellipse, so you're not talking about a harmonic well. You've assumed an inverse square law without knowing it.

Andrew Mason
Homework Helper
Exactly. Your argument fails at any other points. That's cherry-picking.
The argument is more difficult to make at other points. But if you assume that the force is a function of distance from the sun and is radial, all you need are two points (ie. at which you know the speed and the distance from the sun). I don't think Philip has made any implicit assumption about a 1/r^2 force.

By applying Kepler's second law or the conservation of angular momentum (since there is no torque if the force is central), we can conclude that mvr is constant which implies that v is proportional to 1/r. Since at these two points there is no tangential acceleration (since the force is radial and the velocity is perpendicular to the force, there is no tangential component to the force), the acceleration = v^2/r. This implies that the force is mv^2/r.

No, you haven't deduced the inverse square law. First off, you used the only two points where ##v^2/\rho = a## and used these cherry-picked points to make a universal deduction. That's not valid.

Secondly, you've deduced something that you already assumed. There are only two classes of radial central forces that can result in elliptical motion, force proportional to distance and force inversely proportional to distance squared. You've put the object responsible for that central force at one of the foci of the ellipse, so you're not talking about a harmonic well. You've assumed an inverse square law without knowing it.
I would agree if you assume that the orbit is an ellipse. But all one has to assume is that it is a closed loop of some kind, that the force is central and is a function of distance. Then I think that all you need are two points where the tangential acceleration is zero.

AM

D H
Staff Emeritus
By applying Kepler's second law or the conservation of angular momentum (since there is no torque if the force is central), we can conclude that mvr is constant which implies that v is proportional to 1/r.
You don't want Kepler's second law here because that implicitly assumes a 1/r2 force. Conservation of angular momentum would work, but it does not say that mvr is constant. It says that ##m\vec v \times \vec r## is constant. There's a big difference between that vector formulation and the scalar expression mvr=constant. In fact, that scalar expression is not constant for an elliptical orbit.

Since at these two points there is no tangential acceleration (since the force is radial and the velocity is perpendicular to the force, there is no tangential component to the force), the acceleration = v^2/r. This implies that the force is mv^2/r.
The acceleration at these points is v2/ρ, not v2/r. There's a big difference here between ρ (radius or curvature) and r (distance to the central body). The radius of curvature is only equal to the distance to the central body in the case of uniform circular motion.

I would agree if you assume that the orbit is an ellipse. But all one has to assume is that it is a closed loop of some kind, that the force is central and is a function of distance. Then I think that all you need are two points where the tangential acceleration is zero.
Motion in a harmonic potential well provides an immediate counterexample. This yields elliptical motion. (It's the only other simple radial force law other than 1/r2 that yields elliptical motion.) Pick an endpoint of the major axis and an endpoint of the minor axis. You do not get an inverse square law. You instead get a force that is proportional to the distance to the center of the ellipse.

Philip Wood
Gold Member
"You don't want Kepler's second law here because that implicitly assumes a 1/r^2 force."[DH]

It does not. It is consistent with any central force (i.e. any force directed to a fixed point). (Principia Book I Section II Prop1 Theorem1).

"Your argument fails at any other points." [DH]

"Fails" could mean more than one thing. It could mean that if the same argument is applied at other points, then a contradictory conclusion is reached. Presumably this isn't what you mean, because I'm confident that no such conclusion can be drawn. I take it that you mean that the argument can't be used at other points - a far less serious charge, and one which I've never disputed. The argument can't be used at other points because things are more complicated than for the special points A and P. For one thing, A and P are the only points for which $\left|m\vec{r}\times \vec{v}\right| = mrv$, as you yourself point out. I think that the first meaning of "the argument fails" is in danger of contaminating the second, non-pejorative, meaning.

Please consider the limited nature of the claims I'm making. They are these:
(1) $\frac{F_P}{F_A} =\frac{r_A^2}{r_P^2}.$ Do you agree with this?
(2) If the force is a simple power law force directed to one focus, then (1) is sufficient to show that the power law is an inverse square. Just one pair of points suffices, given the assumptions.

I re-iterate: I know that a complete treatment analysis of Kepler's laws I and II, as performed by Newton, doesn't make these assumptions. My argument, as originally presented, has more modest aims.

Andrew Mason
Homework Helper
You don't want Kepler's second law here because that implicitly assumes a 1/r2 force.
Newton proved that a force exerted from a central point on a body moving with an arbitrary velocity relative to the radial line from that central point results in the body's path prescribing equal areas in equal times without any assumption about the nature of the force: (Principia, Book I, Sec. II, Prop. I, Theorem I).

Conservation of angular momentum would work, but it does not say that mvr is constant. It says that ##m\vec v \times \vec r## is constant. There's a big difference between that vector formulation and the scalar expression mvr=constant. In fact, that scalar expression is not constant for an elliptical orbit.
Quite right. L = mv x r is constant, but when r is at an extremum, the velocity vector is perpendicular to the radial vector so mvr is the same at all extrema.

The acceleration at these points is v2/ρ, not v2/r. There's a big difference here between ρ (radius or curvature) and r (distance to the central body). The radius of curvature is only equal to the distance to the central body in the case of uniform circular motion.
There is a component of tangential acceleration in any neighbourhood of the extrema of r (being the distance to the central point) and this component + the component of acceleration toward the instantaneous centre of rotation (v^2/ρ) must sum to v^2/r .

Motion in a harmonic potential well provides an immediate counterexample. This yields elliptical motion. (It's the only other simple radial force law other than 1/r2 that yields elliptical motion.) Pick an endpoint of the major axis and an endpoint of the minor axis. You do not get an inverse square law. You instead get a force that is proportional to the distance to the center of the ellipse.
That is true, but if it was a harmonic potential, ra would be the same as rb. In this case ra ≠ rb.

AM

D H
Staff Emeritus
There is a component of tangential acceleration in any neighbourhood of the extrema of r (being the distance to the central point) and this component + the component of acceleration toward the instantaneous centre of rotation (v^2/ρ) must sum to v^2/r .
That's just wrong, Andrew. Think about it. Acceleration is a continuous function of time (or of true anomaly, or of however else you want to parameterize the motion), and thus so is its magnitude. The magnitude of the acceleration at the extrema of radial position is v^2/ρ, and that is not equal to v^2/r (I'm assuming your r is the distance to the central body). This means there's a neighborhood of those extrema where the magnitude of acceleration is within some epsilon of v^2/ρ, and that v^2/ρ±ε does *not* include v^2/r.

That is true, but if it was a harmonic potential, ra would be the same as rb. In this case ra ≠ rb.
You didn't say I had to look at the two endpoints of the major axis. You said "Then I think that all you need are two points where the tangential acceleration is zero." The choice of points is mine so long as the tangential acceleration is zero at both points. With that harmonic potential I'll pick one of the points as an endpoint of the major axis and the other as an endpoint of the minor axis.

D H
Staff Emeritus
Please consider the limited nature of the claims I'm making. They are these:
(1) $\frac{F_P}{F_A} =\frac{r_A^2}{r_P^2}.$ Do you agree with this?
It depends on what you mean with those suffixes A and P. If you mean the apofocus and perifocus of an ellipse with the central object at one of the foci of the ellipse, well of course. But with those conditions you've implicitly assumed an inverse square force.

If on the other hand you mean more generally the force at the minimum and maximum of radial position, ##\frac{F(r_{\text{min}})} {F(r_{\text{max}})} = \bigl( \frac{r_{\text{max}}} {r_{\text{min}}} \bigr)^2##, then no. This is not true for all radial central forces.

(2) If the force is a simple power law force directed to one focus, then (1) is sufficient to show that the power law is an inverse square. Just one pair of points suffices, given the assumptions.
Again, you've implicitly assumed an inverse square law here. "If the force is a simple power law force directed to one focus" -- there's only one power law that yields this behavior, an inverse square law.

Just by saying "focus" you are implicitly assuming an ellipse. There are only two power laws that yield elliptical motion, radial harmonic and inverse square. It's a consequence of Bertrand's theorem. By putting the central body at a focus you eliminated the radial harmonic case (the central body is at the center of the ellipse rather than a focus). That leaves only an inverse square law. Assuming what you are going to prove and then later deriving what you assumed is not a valid form of proof.