For an object of mass m to topple on a rough inclined plane, we consider the torques due to forces acting on it as shown in this figure:

Is normal reaction a non-central force?

In the figure, we can see that forces along y-axis are cancelling each other. Hence, net torque about the bottom-right point of box would be, $\tau = Wsin\theta \frac{h_{box}}{2}$, which would be non-zero from the beginning of inclination!

But that's not true in practical. So, I thought that "normal reaction" should be a non-central force, which is again not true.

Help me to resolve this paradox.


Remember that the normal force is localized on the plane, not at the center of mass of the cube. The torque from the friction force will try to make the cube rotates clockwise. This will put more pressure on the front the cube, resulting in the normal force to be shifted to the front the cube. The magnitude and the direction of the normal force won't change, but its effective position between the plane and the cube will. In other words, the effective normal force will simply take whatever position between the cube and the plane it must to cancel the torque from the friction force.

Knowing that, you can even compute how big of a friction force you need to make the cube rotate. It's when the effective normal force is applied at the far right edge. Not so surprisingly, since the friction force also depends on the mass of the object, only the geometry of your object, the friction coefficient and the angle of the plane will determine if your object will suddently rotate.

Edit: The friction force relative position will also move with the normal force, but this effect actually decreases the torque from the friction force, which also helps the cube to not rotate.


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