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Imagine a door with 2 hinges. If you place a motor on a hinge, how much force is needed to move the door 90 degrees in X seconds. Also, if you place more motors, 2 for example, will said motors share the force equally?

Thank you.

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The torque applied at a hinge will be $\tau(t) = I\alpha(t)$, where $I$ is the moment of inertia for rotating the door about the hinge and $\alpha$ is the angular acceleration of the door. The torque and the acceleration can be functions of time.

Rearranging, we get $\alpha(t) = \frac{\tau(t)}{I}$ and the angular velocity of the door at time $t$ is the integral $\omega(t) = \int \alpha(t) dt$. Furthermore, the angular velocity is related to the angular position $\theta(t)$ via $\omega(t) = \frac{d}{dt}\theta(t)$ or $\theta(t) = \int \omega(t) dt$.

We'd like to know some value of constant torque, $\tau(t)=\tau_0$ (no time dependence), such that $\theta(t=X)=90˚$.

Expressing angular position as a function of torque, we get $\theta(t)=\int\int \frac{\tau_0}{I} dtdt = \frac{1}{2}\frac{\tau_0}{I}t^2$. Finally, we solve for $\tau_0$ using $\theta(t=X)=\frac{1}{2}\frac{\tau_0}{I}(X)^2=90˚$.

If an additional motor is used, the two torques will add as vectors. So, $\tau_0=\tau_{1,0}+\tau_{2,0}$.

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