# 3.6: Angular Momentum and Torque

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The angular momentum of body, $$dm$$, is defined as $L = r \times U dm$ The angular momentum of the entire system is calculated by integration (summation) of all the particles in the system as $L_{s} = \int_{m} r \times U dm$ The change with time of angular momentum is called torque, in analogous to the momentum change of time which is the force. $T_{\tau} = \frac{DL}{Dt} = \frac{D}{Dt}\left(r \times U dm\right)$ where $$T_{\tau}$$ is the torque. The torque of entire system is $T_{\tau s} = \int_{m} \frac{DL}{Dt} = \frac{D}{Dt} \int_{m} \left(r \times U dm \right)$ It can be noticed (well, it can be proved utilizing vector mechanics) that $T_{\tau} = \frac{D}{Dt}\left(r \times U \right) = \frac{D}{Dt}\left(r \times \frac{Dr}{Dt} \right) = \frac{D^{2}r}{Dt^{2}}$ To understand these equations a bit better, consider a particle moving in x–y plane. A force is acting on the particle in the same plane (x–y) plane. The velocity can be written as $$U = u \hat{i} + v\hat{j}$$ and the location from the origin can be written as $$r = x \hat{i} + y \hat{j}$$. The force can be written, in the same fashion, as $$F = F_{x} \hat{i} + F_{y} \hat{j}$$. Utilizing equation 61 provides $matrix$ Utilizing equation 63 to calculate the torque as $matrix$ Since the torque is a derivative with respect to the time of the angular momentum it is also can be written as $xF_{x} - yF_{y} = \frac{D}{Dt}\left[\left(xv - yu\right)dm\right]$ The torque is a vector and the various components can be represented as $T_{\tau x} = \hat{i} \cdot \frac{D}{Dt} \int_{m} r \times U dm$ In the same way the component in $$y$$ and $$z$$ can be obtained.