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