12. “The farther away, the less it should weigh.” Express this in an equation of spinal torque, considering object and body weight.

1. Break Down the Problem

We need to establish a relationship between the torque exerted by an object's weight and the distance from the pivot (in this case, the spine). The statement implies that as the distance from the body (pivot point) increases, the effective weight (or torque) should decrease.

2. Relevant Concepts

Torque ($\tau$) can be expressed as:

$\tau = r \times F$

where:

• $r$ = distance from the pivot point (spine) to the point of force application (where the object is held)
• $F$ = force due to weight of the object, which is equal to the mass ($m$) of the object times the gravitational acceleration ($g$), i.e., $F = m \cdot g$

3. Analysis and Detail

From the statement "The farther away, the less it should weigh," we can replace $F$ with $m \cdot g$ in the torque equation:

$\tau = r \cdot (m \cdot g)$

However, if we interpret the statement literally, it could imply that effective torque perceived decreases with distance:

$\tau_{effective} = \frac{m \cdot g}{r}$

This indicates that as $r$ (the distance from the pivot) increases, the effective torque perceived on the body decreases.

4. Verify and Summarize

Thus, we have established the relationship:

$\tau_{effective} \propto \frac{1}{r}$

This means as the distance $r$ increases, $\tau_{effective}$ (the effective weight when considering torque) decreases.

Final Answer

The equation expressing the idea that "the farther away, the less it should weigh" in terms of spinal torque can be represented as:

$\tau_{effective} = \frac{m \cdot g}{r}$