Main variables:
\( \vec{a} \) = Acceleration.
\( \vec{F} \) = Force.
\( m \) = Mass.
\( g \) = Gravitational acceleration.
\(\vec{W} \) = Weight.
\( \vec{N} \) = Normal force.
\( \vec{f}_s \) = Static friction force.
\( \vec{f}_k \) = Kinetic friction force.
\( \mu_s \) = Static friction coefficient.
\( \mu_k \) = Kinetic friction coefficient.
\( v \) = Tangential speed for a circular motion.
\( R \) = Radius of a circular motion.
\(T \) = Period for a circular motion.
\( \hat{\textbf{i}} \) = Unitary vector along X-axis.
\( \hat{\textbf{j}} \) = Unitary vector along Y-axis.
\( \sum F\) = Sum over all the forces.
\( a \le b \) = Reads “\(a\) less or equal than \(b\)”, which means \(a\) could take all the possible values less than \(b\) but also can be equal to \(b\).
When we write \(a\) instead of \(\vec{a}\), we refer to the magnitude of that vector. For instance, \(\vec{a}\) is the acceleration vector, and \(a\) is the magnitude of the acceleration.
Main Equations:
Newton’s second law:
\begin{equation*}
\sum \vec{F} = m \vec{a}.
\end{equation*}
When the object is at equilibrium (or constant velocity):
\begin{equation*}
\sum \vec{F} = 0.
\end{equation*}
The weight of an object near the Earth’s surface:
\begin{equation*}
\vec{W} = m \vec{g}.
\end{equation*}
The static friction force:
\begin{equation*}
\vec{f}_{s} \le \mu_s \vec{N},
\end{equation*}
The maximum static friction force:
\begin{equation*}
\vec{f}_{s}^{\text{ max}} = \mu_s \vec{N}.
\end{equation*}
The kinetic friction force is:
\begin{equation*}
\vec{f}_{k} = \mu_k \vec{N}.
\end{equation*}
For uniform circular motion, the magnitude of radial acceleration:
\begin{equation*}
a_{rad} = \frac{v^2}{R},
\end{equation*}
and the direction of this acceleration points towards the center of the circle.
The speed of a circular motion:
\begin{equation*}
v = \frac{2 \pi R}{T}.
\end{equation*}