Editing Circular motion (section)

== Applications ==
Solving applications dealing with non-uniform circular motion involves force analysis. With a uniform circular motion, the only force acting upon an object traveling in a circle is the centripetal force. In a non-uniform circular motion, there are additional forces acting on the object due to a non-zero tangential acceleration. Although there are additional forces acting upon the object, the sum of all the forces acting on the object will have to be equal to the centripetal force.
<math display="block">\begin{align}
  F_\text{net} &= ma \\


               &= ma_r \\
               &= \frac{mv^2}{r} \\
               &= F_c
\end{align}</math>

Radial acceleration is used when calculating the total force. Tangential acceleration is not used in calculating total force because it is not responsible for keeping the object in a circular path. The only acceleration responsible for keeping an object moving in a circle is the radial acceleration. Since the sum of all forces is the centripetal force, drawing centripetal force into a free body diagram is not necessary and usually not recommended.

Using <math>F_\text{net} = F_c</math>, we can draw free body diagrams to list all the forces acting on an object and then set it equal to <math>F_c</math>. Afterward, we can solve for whatever is unknown (this can be mass, velocity, radius of curvature, coefficient of friction, normal force, etc.). For example, the visual above showing an object at the top of a semicircle would be expressed as <math>F_c = n + mg</math>.

In a uniform circular motion, the total acceleration of an object in a circular path is equal to the radial acceleration. Due to the presence of tangential acceleration in a non uniform circular motion, that does not hold true any more. To find the total acceleration of an object in a non uniform circular, find the vector sum of the tangential acceleration and the radial acceleration.
<math display="block">\sqrt{a_r^2 + a_t^2} = a</math>

Radial acceleration is still equal to <math display="inline">\frac{v^2}{r}</math>. Tangential acceleration is simply the derivative of the speed at any given point: <math display="inline">a_t = \frac{dv}{dt} </math>. This root sum of squares of separate radial and tangential accelerations is only correct for circular motion; for general motion within a plane with polar coordinates <math>(r, \theta)</math>, the Coriolis term <math display="inline">a_c = 2 \left(\frac{dr}{dt}\right)\left(\frac{d\theta}{dt}\right)</math> should be added to <math>a_t</math>, whereas radial acceleration then becomes <math display="inline">a_r = \frac{-v^2}{r} + \frac{d^2 r}{dt^2}</math>.