| \(F = ma\) |
\(\vec{F} = M \vec{a}\) |
The force is collinear to the acceleration and is proportional to the mass. |
| \(p = mv\) |
\(\vec{p} = M \vec{v}\) |
The momentum is collinear to the velocity and is proportional to the mass. |
| \(\tau = I \alpha\) |
\(\vec{\tau} = I \vec{\alpha} + \vec{\omega} \times I \vec{\omega}\) |
The vector form in 3D is the more accurate representation. It takes into account the fact that there can be a torque even without an angular acceleration, just because of the shape of the object. Check Moment of Intertia for more information. The torque might not be collinear with the angular acceleration. |
| \(\tau = rF\) |
\(\vec{\tau} = \vec{r} \times \vec{F}\) |
The cross product handles situations where the radius is not perpendicular to the force. |
| \(L = I\omega\) |
\(\vec{L} = I \vec{\omega}\) |
The angular momentum need not be collinear with the angular velocity. Check Moment of Intertia for more information. |
| \(v_t = \omega r\) |
\(\vec{v_t} = \vec{\omega} \times \vec{r}\) |
The cross product handles situations where the radius is not perpendicular to the angular velocity. The resulting tangential velocity, when not zero, is perpendicular to the angular velocity. |
| \(a_t = \alpha r\) |
\(\vec{a_t} = \vec{\alpha} \times \vec{r}\) |
The cross product handles situations where the radius is not perpendicular to the angular acceleration. |
| \(F = \mu F_n\) |
\(\vec{F_{max}} = - \mu \hat{v_{t}} \|\vec{F_n}\|\) |
This returns the maximum friction force. The actual force could be less if it would be sufficient to keep the object from moving. |
| \(F_d = \frac{1}{2} \rho v^2 C_d A\) |
\(\vec{F_d} = \frac{1}{2} \rho C_d A \|\vec{v}\| \vec{v}\) |
The velocity in this equation is the relative velocity of the wind to the object. |
| \(F_l = \frac{1}{2} \rho v^2 C_L A\) |
\(\vec{F_l} = \frac{1}{2} \rho C_L A (\vec{v} \cdot \vec{v}) \hat{n}\) |
The velocity in this equation is the relative velocity of the wind to the object. The \(\hat{n}\) term defines the lift direction of the wing, perpendicular to the wind direction. |
| \(F_s = k x\) |
\(\vec{F_s} = -k \vec{x}\) |
This is according to Hooke's law, though not all springs follow that law. The force is in the opposite direction to the displacement. |