Substituting this to the equation (2), the equation (5) can be obtained. The acceleration, ''$\alpha$'' can be obtained by differentiating the equation (3) further with respect to the time as shown in the equation (4). In classical, non-relativistic physics, it is a scalar quantity (often denoted by the symbol ) and, like length, mass, and charge, is usually described as a fundamental quantity. In the firure above, the speed, ''$v$'' at the radius of gyration assuming that the mass, ''$m$'' of the balance wheel converged at a point, can be obtained by differentiating the moving distance, ''$rd\theta$'' with respect to the time, as shown in the equation (3) which is the same as that of the pendulum. en. wiki TheoryofrelativityTheory of relativity - Wikipedia Time in physics is defined by its measurement: time is what a clock reads. As ''$F_r$'' represents '' Force $\times$ distance (Force by distance)'', this represents the torque, ''$T$''. Multiplying the distance, ''$r$'' at the both sides of the equation (1), the equation (2) can be obtained. Now, deriving the equation of motion of the balance, back to the Newton's law, equation (1). In case of the pendulum and the balance, the restoring force is accordingly gravity and the spring constant of the hairspring respectively. ''$k$'' of this equation is called as ''spring constant''. In this situation, the equation of ''$T$ (the restoring torque)$ = - k\theta$'' can be derived. This restoring torque is in proportion to the rotation angle, ''$\theta$'' degrees of the balance wheel. At this moment, the hairspring also intends to return back to the previous position as well, the restoring torque is also generated at the balance wheel. Inversely, when the balance wheel rotates clockwise direction, the hairspring will change the shape so that the radius of curvature can be got smaller. Then, as the hairspring intends to return back to the previous position, the restoring torque is generated at the balance wheel. In terms of the rotational direction of the balance wheel, deciding the ''+'' for the counterclockwise, the hairspring will change the shape so that the radius of curvature can be enlarged. The figure below shows the situation that the balance wheel is rotated by ''$\theta$'' degrees in the counterclockwise. The outer end of the hairspring is fixed. This is based on the mechanism that makes the reciprocating rotational motion as well as the pendulum by fixing the mass which is the balance wheel in the center part of the spring called as the hairspring.
The wrist watch makes the oscillation of the constant period using the balance in place of the pendulum.
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