The Directorate moment , also the restoring torque or target torque is at a mechanical twist the constant of proportionality between the applied torque and the rotation angle :
![{\ displaystyle {\ vec {\ varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f82bcb624dfe10df9c55ed401628ed7ff7a02f3)
![{\ displaystyle {\ vec {M}} = D \ cdot {\ vec {\ varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3387ad0b18a3ec7186b80ab321521dd3bfc0be5d)
The directional moment corresponds to the spring constant for longitudinal deflections.
In a system capable of oscillation, for example the balance wheel of a watch, the period of oscillation and the moment of inertia can be used to calculate:
![J](https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053)
![{\ displaystyle D = 4 \ pi ^ {2} \ cdot {\ frac {J} {T ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd73584dfc503bac2205cfcfe58170fd5cc9851b)
For circular cylindrical wires with the radius and length , the directional moment is proportional to the shear modulus :
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle D = {\ frac {\ pi \ cdot r ^ {4}} {2}} \ cdot {\ frac {G} {l}} = {\ frac {I_ {p} \ cdot G} {l }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/479853c0afcb60d5232fc93ad29be7f990842c72)
with the polar area moment of inertia .
![I_p](https://wikimedia.org/api/rest_v1/media/math/render/svg/3226e9ad60e659391806720213f9c5f6123a70f8)
For example, for a steel wire with the values and you get .
![{\ displaystyle G _ {\ mathrm {Fe}} = 80 \, \ mathrm {GPa}, l = 0.1 \, \ mathrm {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9473217ecac140d788fc6232913ea41c80e5a1)
![{\ displaystyle r = 2.5 \, \ mathrm {mm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8770681db255f55ae5dba7b76cb54fbc66b503)
![{\ displaystyle D \ approx 50 \, \ mathrm {Nm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d799925bbd0cdddf2ad28ba8f529cecb33a228)
A torque of twists it around:
![{\ displaystyle M = 1 \, \ mathrm {Nm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fafcfab30e33b418f44380d95cf22bfcb23d42)
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(rad = radian )
See also