The equation shown in the image is:

$\tau = \frac{T}{G_o J_o + G_i J_i} \times G \times \mathbf{r}$

To help you derive this, I need some context about the physical system or domain this equation comes from. Based on the symbols used, it looks like a torque ($\tau$) related formula involving:

• $T$ – possibly torque or thrust.
• $G_o$ and $G_i$ – could be gear ratios or constants.
• $J_o$ and $J_i$ – likely represent moments of inertia (possibly for output and input, respectively).
• $G$ – might be a gear ratio.
• $\mathbf{r}$ – a position or radius vector.

This appears to involve the distribution of torque in a rotational system, possibly involving coupled gears or rotating masses.

Without additional context, here’s a general derivation approach:

1. Moment of Inertia Contributions: The terms $J_o$ and $J_i$ likely refer to moments of inertia of two different objects or components (e.g., input and output shafts in a gearbox). In rotational dynamics, torque $\tau$ is related to the angular acceleration $\alpha$ by: $\tau = J \alpha$ Here, $J$ is the moment of inertia, and if there are multiple objects, their combined effect is additive: $\tau = (J_o + J_i) \alpha$

2. Gear Ratios: Gear ratios $G_o$ and $G_i$ are often used to translate between angular velocities or torques between two linked rotational systems. The gear ratio modifies the effective moment of inertia seen by each shaft:

   \quad \text{and} \quad \tau_i = G_i J_i \alpha_i$$ Depending on how the gear system is arranged, these expressions can be combined for total torque.

3. Combining Torques: The total torque $T$ would likely be split across the different components, thus the equation might represent how torque is distributed based on the moment of inertia and gear ratios of the system. The denominator $G_o J_o + G_i J_i$ represents the total "effective inertia" of the system.

4. Radius Vector $r$: The final factor $\mathbf{r}$ could account for a distance in the system, like a radial arm or distance from the axis of rotation, converting angular torque into a linear force.

Can you confirm the context? Then I can assist more precisely in the derivation.

Would you like more detailed steps on any part?

Here are 5 questions related to the formula:

1. What is the specific physical system (gears, rotational body) this formula applies to?
2. What does each symbol (especially $G_o$, $G_i$, $J_o$, and $J_i$) represent in your system?
3. Does $T$ represent total torque or another quantity like thrust?
4. Is $\mathbf{r}$ a radial distance or a vector related to position?
5. How do $G_o$ and $G_i$ interact in your system (input-output relationship)?

Tip: Always verify the physical meaning of each symbol to properly understand how to derive equations!