Main variables:
\( \vec{F} \) = Force.
\( \vec{x} \) = Position vector.
\(W \) = Work.
\(m \) = Mass.
\(t\) = Time.
\(\vec{v} \) = Velocity.
\( K \) = Kinetic energy.
\( P \) = Power.
\( U_g \) = Gravitational potential energy.
\( U_{el} \) = Elastic potential energy.
\(h \) = Height of an object.
\(k \) = Spring constant.
\( \nabla \) = Nabla operator or gradient when its applied to a scalar field.
\( W_{\text{other}} \) = Work by non-conservative forces.
Subindex ‘i’ means initial. Subindex ‘f’ means final. For example, \(\vec{v}_i\) represents the initial velocity.
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:
The dot product between two vectors:
\begin{equation*}
\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y,
\end{equation*}
or
\begin{equation*}
\vec{A} \cdot \vec{B} = A B \cos \theta,
\end{equation*}
where \( \theta \) is the angle between the two vectors.
The work done by a force:
\begin{equation*}
W = \vec{F} \cdot \vec{x}.
\end{equation*}
Work when \( \vec{F} \) and \(\vec{x}\) are perpendicular:
\begin{equation*}
W = 0.
\end{equation*}
Work when \( \vec{F} \) and \(\vec{x}\) are parallel:
\begin{equation*}
W = Fx.
\end{equation*}
Work when \( \vec{F} \) and \(\vec{x}\) are antiparallel:
\begin{equation*}
W = -Fx.
\end{equation*}
The work by a variable force:
\begin{equation*}
W = \int_{x_1}^{x_2} \vec{F}(x) \cdot d\vec{x}.
\end{equation*}
Kinetic energy:
\begin{equation*}
K = \frac{1}{2} m v^2.
\end{equation*}
The work-energy theorem is:
\begin{equation*}
W = \Delta K = K_f – K_i.
\end{equation*}
The power in terms of the work:
\begin{equation*}
P = \frac{\Delta W}{\Delta t}.
\end{equation*}
The power as a differential form in terms of the work:
\begin{equation*}
P = \frac{ dW}{dt}.
\end{equation*}
The power in terms of the force:
\begin{equation*}
P = \vec{F} \cdot \vec{v}.
\end{equation*}
The gravitational potential energy:
\begin{equation*}
U_{g} = mgh.
\end{equation*}
The elastic potential energy in terms of \(x\) the distance between the equilibrium position:
\begin{equation*}
U_{el} = \frac{1}{2} kx^2.
\end{equation*}
Conservation of mechanical energy:
\begin{equation*}
K_1 + U_{g_1} + U_{el_1} = K_2 + U_{g_2} + U_{el_2}.
\end{equation*}
When the mechanical energy is NOT conserved:
\begin{equation*}
K_1 + U_{g_1} + U_{el_1} + W_{other} = K_2 + U_{g_2} + U_{el_2}.
\end{equation*}
When the force is conservative (it conserves the mechanical energy of the system), we can write it in terms of the potential energy in this manner:
\begin{equation*}
\vec{F} = – \nabla U = – \left( \frac{\partial U}{\partial x} \hat{\textbf{i}} + \frac{\partial U}{\partial y} \hat{\textbf{j}} + \frac{\partial U}{\partial z} \hat{\textbf{k}} \right).
\end{equation*}