Really confused by why centripetal acceleration can have two components

Please see the attached scan from my Mechanics 3 textbook.

I am completely happy up to the point which shows that the v vector and r vector are perpendicular, thus confirming that velocity in uniform circular motion is always perp. to radius vector.

However, I am confused where I have placed the marker: how can the acceleration vector, which by definition is always centripetal, have a non zero component along the tangent of the circle at that point, ie same direction as v vector?? As we all know, two perp. vectors have 0 component in direction of other vector?

All i can think that is myabe because we are dealing with vertical circles, the weight of the particle come sinto play, but then we have derived the result son that page from just position vector r, so surely weight/mass is not relevant?

Obviously I am very confused by this!!

Thank you in advance.

UCM vertical circle acc. components.pdf240.2KB

Reply 1

ghostwalker

Original post by Choochoo_baloo

This is true of all circular motion which is what you're looking at here, not just uniform circular motion.

However, I am confused where I have placed the marker: how can the acceleration vector, which by definition is always centripetal, have a non zero component along the tangent of the circle at that point, ie same direction as v vector?? As we all know, two perp. vectors have 0 component in direction of other vector?

What definition are you refering to?

Only if the speed in a circle is constant is the acceleration soley centripetal.

All i can think that is myabe because we are dealing with vertical circles, the weight of the particle come sinto play, but then we have derived the result son that page from just position vector r, so surely weight/mass is not relevant?

This is a general derivation applying to all motions in a circle.

Reply 2

atsruser

Original post by Choochoo_baloo

This is not specifically related to motion in a vertical circle. The result is true for any circular motion where the particle is not moving with a constant speed around the circle. In that case, the acceleration vector has two components, one of which is centripetal,

r\omega^2

, and another which is tangential,

r\ddot{\theta}

.

Since the speed of the particle varies as it moves round the circle, there must be a component of acceleration in the direction of its velocity (i.e. tangential to the circle), else the magnitude of velocity vector could not change.

In addition, the centripetal acceleration is not constant, since it is

r\dot{\theta}^2

, and hence changes if

\dot{\theta}

changes. But of course,

\dot{\theta}

*will* change, since

\ddot{\theta}

is non-zero (hence its time-integral, which is

\dot{\theta}

, cannot be constant).

Note that in the case of uniform circular motion, the centripetal acceleration is

r\dot{\theta}^2

, but in this case,

\dot{\theta}

is constant. In the case of non-uniform motion, the centripetal acceleration is still

r\dot{\theta}^2

, but its magnitude varies depending upon how fast the particle is moving around the circle at any particular instant.

Note that since the acceleration now has two components, one centripetal and one tangential. the acceleration vector doesn't, in general point towards the centre of the circle. Its precise direction depends on the relative sizes of the two components.