Editing Circular motion (section)

== Non-uniform circular motion ==
[[File:Nonuniform circular motion.svg|220x220px|thumb|Velocity and acceleration in non-uniform circular motion.]]
In '''non-uniform circular motion''', an object moves in a circular path with varying [[speed]]. Since the speed is changing, there is [[tangential acceleration]] in addition to normal acceleration.

The net acceleration is directed towards the interior of the circle (but does not pass through its center). 

The net acceleration may be resolved into two components: tangential acceleration and centripetal acceleration. Unlike tangential acceleration, centripetal acceleration is present in both uniform and non-uniform circular motion.

[[File:Freebody circular.svg|left|thumb|220x220px|This diagram shows the normal force (n) pointing in other directions rather than opposite to the weight force.]]
In non-uniform circular motion, the [[normal force]] does not always point to the opposite direction of [[weight]].

[[File:Freebody object.svg|thumb|220x220px|Here, 'n' is the normal force.]]
The normal force is actually the sum of the radial and tangential forces. The component of weight force is responsible for the tangential force (when we neglect friction). The centripetal force is due to the change in the direction of velocity.

The normal force and weight may also point in the same direction. Both forces can point downwards, yet the object will remain in a circular path without falling down. 

[[File:Normal and weight.svg|left|thumb|220x220px|The normal force can point downwards.]]
The normal force ''can'' point downwards. Considering that the object is a person sitting inside a plane moving in a circle, the two forces (weight and normal force) will point down only when the plane reaches the top of the circle. The reason for this is that the normal force is the sum of the tangential force and centripetal force. The tangential force is zero at the top (as no work is performed when the motion is perpendicular to the direction of force). Since weight is perpendicular to the direction of motion of the object at the top of the circle and the centripetal force points downwards, the normal force will point down as well. 

From a logical standpoint, a person travelling in that plane will be upside down at the top of the circle. At that moment, the person's seat is actually pushing down on the person, which is the normal force.

The reason why an object does not fall down when subjected to only downward forces is a simple one. Once an object is thrown into the air, there is only the downward gravitational force that acts on the object. That does not mean that once an object is thrown into the air, it will fall instantly. The [[velocity]] of the object keeps it up in the air. The first of [[Newton's laws of motion]] states that an object's [[inertia]] keeps it in motion; since the object in the air has a velocity, it will tend to keep moving in that direction.

A varying [[angular speed]] for an object moving in a circular path can also be achieved if the rotating body does not have a homogeneous mass distribution.<ref>{{cite journal |last1=Gomez |first1=R W |last2=Hernandez-Gomez |first2=J J |last3=Marquina |first3=V |date=25 July 2012 |title=A jumping cylinder on an inclined plane |url=https://www.researchgate.net/publication/236030807 |journal=Eur. J. Phys. |publisher=IOP |volume=33 |issue=5 |pages=1359–1365 |arxiv=1204.0600 |bibcode=2012EJPh...33.1359G |doi=10.1088/0143-0807/33/5/1359 |s2cid=55442794 |access-date=25 April 2016}}</ref>

One can deduce the formulae of speed, acceleration and jerk, assuming that all the variables depend on <math> t </math>:
<math display="block">
\mathbf{r} = R \mathbf{u}_R
</math>
<math display="block">
\dot \mathbf{u}_R = \omega\mathbf{u}_{\theta}
</math>
<math display="block">
\dot \mathbf{u}_{\theta} = -\omega\mathbf{u}_R
</math>
<math display="block">
\mathbf{v} = \frac{d}{dt} \mathbf{r} = \dot \mathbf{r}
= \dot R \mathbf{u}_R + R \omega \mathbf{u}_{\theta}
</math>
<math display="block">
\mathbf{a} = \frac{d}{dt} \mathbf{v} =  \dot \mathbf{v}
= \ddot R \mathbf{u}_R
+ \left(\dot R \omega \mathbf{u}_{\theta}
+ \dot R \omega \mathbf{u}_{\theta} \right)
+ R \dot \omega \mathbf{u}_{\theta}
- R  \omega^2 \mathbf{u}_{R} 
</math>

<math display="block">
\mathbf{j} = \frac{d}{dt} \mathbf{a} = \dot \mathbf{a}
= \dot\ddot R \mathbf{u}_R 
+ \ddot R \omega \mathbf{u}_{\theta}

+ \left(2\ddot R \omega \mathbf{u}_{\theta}
+ 2\dot R \dot\omega \mathbf{u}_{\theta}
- 2\dot R \omega^2\mathbf{u}_R \right)

+ \dot R \dot\omega \mathbf{u}_{\theta}
+ R \ddot\omega \mathbf{u}_{\theta}
- R \dot\omega \omega\mathbf{u}_{R}

- \dot R  \omega^2 \mathbf{u}_{R}
- R  2\dot\omega\omega \mathbf{u}_{R}
- R  \omega^3 \mathbf{u}_{\theta}
</math>

<math display="block">
\mathbf{j}
= \left( \dot\ddot R - 3\dot R \omega^2 - 3R \dot\omega \omega \right) \mathbf{u}_R 
+ \left( 3 \ddot R \omega  + 3\dot R \dot\omega + R \ddot \omega - R  \omega^3 \right) \mathbf{u}_{\theta} 
</math>

Further transformations may involve <math>curvature = c = \frac{1}{R}, \omega = \frac{v}{R} = vc </math> and their corresponding derivatives:
<math display="block">\begin{align}
\dot R &= -\frac{\dot c}{c^2} \\
\ddot R &= \frac{2\left(\dot c\right)^2}{c^3} - \frac{\ddot c}{c^2} \\
\dot \omega &= \frac{\dot v R - \dot R v}{R^2} = \dot v c + v \dot c.\\
\end{align}</math>