Work

$W$ $\mathrm{J,\, kg\,m^2/s^2}$ work
$F$ $\mathrm{N}$ force
$F_G$ $\mathrm{N}$ weight force
$F_B$ $\mathrm{N}$ accelerating force
$F_N$ $\mathrm{N}$ normal force
$F_R$ $\mathrm{N}$ friction force
$F_A$ $\mathrm{N}$ external force
$s$ $\mathrm{m}$ distance, way travelled
$h$ $\mathrm{m}$ height
$\gamma$ $\mathrm{rad}$ angle
$a$ $\mathrm{m/s^2}$ acceleration
$g$ $\mathrm{m/s^2}$ gravitational acceleration
$v$ $\mathrm{m/s}$ velocity
$k$ $\mathrm{N/m}$ spring constant
$\varphi$ $\mathrm{rad}$ angle of rotation
$\omega$ $\mathrm{s^{-1}}$ angular velocity
$\alpha$ $\mathrm{s^{-2}}$ angular acceleration
$J$ $\mathrm{kg\,m^2}$ moment of inertia
$r$ $\mathrm{m}$ distance to center of gravity
$f$ $\mathrm{Nm^2/kg^2}$ gravitational constant

General

$$ W = F \, s \, \cos\gamma $$

Special case:
$$ W = F \, s $$

$$ W = \int\limits_{s_1}^{s_2} \! F \, \cos \gamma \; \mathrm{d}s \qquad
W = \int\limits_{s_1}^{s_2} \! \vec{F}(s) \; \mathrm{d}s $$

Lifting Work

$$ W_{HUB} = – \int \! F_G \; \mathrm{d}s $$

$$ W_{HUB} = F_G \, \Delta s \, \cos\gamma \qquad
W_{HUB} = – m \, g \, \Delta s \, \cos\gamma $$

Conditions: $\gamma = \pi = 180\mathrm{deg} $

$$ W_{HUB} = m \, g \, \Delta h $$

Accelerating Work

$$ W_B = m \, a \, s \qquad
W_B = \frac{m}{2} \, v^2 \qquad
W_B = \frac{m}{2} \, \left( v^2 – v_0^2 \right) $$

$$ W_B = – \int \! \vec{F}_B \;\mathrm{d}s \qquad
W_B = \vec{F}_B \, s $$

$$ W_B = m \int \! a \;\mathrm{d}s \qquad
W_B = m \int \frac{\mathrm{d}\vec{v}}{\mathrm{d}t} \;\mathrm{d}s $$

$$ W_B = m \int \frac{\mathrm{d}\vec{s}}{\mathrm{d}t} \;\mathrm{d}v \qquad
W_B = m \int \! \vec{v} \;\mathrm{d}\vec{v} $$

Frictional Work

$$ W_R = F_R \, s \qquad
W_R = \mu\, F_N \, s \qquad
W_R = \mu\, F_G \, s \, \cos\gamma $$

$$ W_R = \int \! \vec{F}_R \; \mathrm{d}\vec{s} $$

Conditions: $\gamma = 0 $

$$ W_R = \mu \, m \, g \, s $$

Spring Tension Work

$$ W_F = \frac{1}{2} \, k \, s^2 \qquad
W_F = \int \! \vec{F} \;\mathrm{d}s \qquad
W_F = \int\limits_{s_{min}}^{s_{max}} \! k \, s \;\mathrm{d}s $$

Rotation, torsion:

$$ W_F = \frac{1}{2} \, k \, \varphi^2 \qquad
W_F = \frac{1}{2} \, k \, \left( \varphi_2^2 – \varphi_1^2 \right) $$

Rotational Work

$$ W_{ROT} = \int\limits_{\varphi_0}^{\varphi_1} \! M(\varphi) \;\mathrm{d}\varphi $$

$$ W_{ROT} = \int \! J \, \alpha \;\mathrm{d}s \qquad
W_{ROT} = \int \! J \, \frac{\mathrm{d}\omega}{\mathrm{d}t} \;\mathrm{d}s \qquad
W_{ROT} = J \int \frac{\mathrm{d}s}{\mathrm{d}t} \;\mathrm{d}\omega $$

$$ W_{ROT} = J \int \! \omega \;\mathrm{d}\omega \qquad
W_{ROT} = \frac{1}{2} \, J \, \omega^2 \qquad
W_{ROT} = \frac{1}{2} \, J \, \left( \omega_1^2 – \omega_0^2 \right) $$

Gravitational Work

Gravitational work is executed when the smaller mass $m_2$ is lifted from $A$ to $B$ (lifting work).

$$ W_{AB} = \int\limits_{r_1}^{r_2} \! F_A \;\mathrm{d}r \qquad
W_{AB} = – \int\limits_{r_1}^{r_2} \! F_G \;\mathrm{d}r $$

$$ W_{AB} = \int\limits_{r_1}^{r_2} \! f \, m_1 \, m_2 \, \frac{1}{r^2} \;\mathrm{d}r \qquad
W_{AB} = f \, m_1 \, m_2 \, \left( \frac{1}{r} – \frac{1}{r} \right) $$

Gravitational constant:
$$ f = 6.673 \cdot 10^{-11} \,\frac{N\,m^2}{kg^2} $$