Q.The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all the torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point?

Concept: Relationship between torque about different points when net force is non-zero.

The torque $\vec{\tau}$ about a point depends on both the forces and their points of application. When we shift the reference point from $O$ to $O'$, the torque about $O'$ is related to the torque about $O$ by:

where $\vec{r}_{OO'}$ is the position vector from $O'$ to $O$, and $\vec{F}_{\text{net}}$ is the vector sum of all forces.

If $\vec{\tau}_O = 0$ but $\vec{F}_{\text{net}} \neq 0$, then:

$\vec{\tau}_{O'} = \vec{r}_{OO'} \times \vec{F}_{\text{net}}$ …

The heart of this question lies in understanding how torque transforms when you change your reference point. Torque is not an intrinsic property of a force system alone—it depends on where you measure it from. When the net force is zero, torque becomes independent of the reference point (a pure couple). But when the net force is non-zero, shifting your reference point changes the torque.

Let's see why this happens mathematically and physically.

Why torque depends on the reference point

Torque measures the turning effect of forces about a chosen point. For a force $\vec{F}_i$ acting at position $\vec{r}_i$ (measured from point $O$), the torque is:

The total torque about $O$ is:

Now suppose we measure torques about a different point $O'$, displaced from $O$ by vector $\vec{a}$. The position of the same force application point relative to $O'$ is $\vec{r}_i' = \vec{r}_i - \vec{a}$.

The torque about $O'$ becomes:

Expanding:

This is the key relationship. The torque about a new point equals the old torque minus the cross product of the displacement with the net force.

Applying to our problem

1. Given conditions: We have $\vec{F}_{\text{net}} = \sum_i \vec{F}_i \neq \vec{0}$ and $\vec{\tau}_O = \vec{0}$ about some particular point $O$.

2. Torque about an arbitrary point $O'$: Using the transformation formula:

3. Is this zero? The cross product $\vec{a} \times \vec{F}_{\text{net}}$ is zero only if:
   • $\vec{a} = \vec{0}$ (we haven't moved; $O' = O$), or
   • $\vec{a}$ is parallel to $\vec{F}_{\text{net}}$ …

Concept: How Torque Transforms When the Reference Point Shifts

For a system of forces with resultant $\vec F_{net} = \sum_i \vec F_i$, shifting the reference point from $O$ to a point displaced by $\vec a$ changes the total torque as

Step 1: Apply the given conditions

We are told $\vec F_{net} \neq \vec 0$ (non-collinear forces, non-zero vector sum) and $\vec\tau_O = \vec 0$ about some particular point $O$.

Step 2: Torque about an arbitrary point $O'$

Step 3: When is this zero?

The cross product $\vec a \times \vec F_{net}$ vanishes only if $\vec a = \vec 0$ (i.e. $O'=O$) or $\vec a$ is parallel to $\vec F_{net}$ — meaning $O'$ lies on the line of action of the resultant force through $O$. For a general arbitrary point, $\vec a$ is not parallel to $\vec F_{net}$, so $\vec\tau_{O'} \neq \vec 0$. …