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I'm reading a book on physics in which there is a line that I do not understand.

I quote it here:

If a body is placed on a horizontal surface, the torque of the contact forces about the centre of mass should be zero to maintain the equilibrium. This may happen only if the vertical line through the centre of mass cuts the base surface at a point within the contact area or the area bounded by the contact points.

Why does this happen? is there any proof?

A similar question has been answered here: Torque of the contact forces about the centre of mass

The answer there concerns why a person has to lean opposite to the direction in which they have a load in one hand.

But here I seek a reason, intuition, or a proof if possible, for the quoted text.

Qmechanic
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Apoorva Shukla
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2 Answers2

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For intuition think about it this way ...

The set of contact forces can be reduced to a single effective point force acting at some point within the contact area (see caveat about convexity of contact area below). It can be proven that this must occur if all of the contact forces are normal to the contact plane (assuming no friction which is reasonable for an object resting on a horizontal surface) and if the contact area is convex. So if the contact area is not convex, then the statement in your book should be modified to account for that contingency. For example, if the contact area was a half-ring, then the effective point of application of the contact force would reside in the area bounded by the semi-circle of associated with the half ring (see picture below). The effective point of application of the point force can only be outside of the contact area if the set of forces can have opposite directions. Perhaps you have this intuition already that the net contact force acts at some point within the contact area.

Once you have that intuition, the next step is to consider the overall equilibrium of the object. There is the contact force upward and the weight acting downward. Balance of forces tells us that the magnitude of the net contact force must be equal to the weight, and balance of moments (torques) tells us that the lines of action of the weight and the net contact force must be identical because if they were offset from one another then there would be a net moment on the object and it would not be in static equilibrium. Hence, the downward force of the weight must point directly through the effective point of application of the contact forces which must be at some location within the contact area. enter image description here

looksquirrel101
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If a body is placed on a horizontal surface, the torque of the contact forces about the centre of mass should be zero to maintain the equilibrium.

Do you understand this part? A non-zero sum would imply a change in angular momentum, which would require either that the object is accelerating or changing its rotation. Neither are compatible with an object at rest.

This may happen only if the vertical line through the centre of mass cuts the base surface at a point within the contact area or the area bounded by the contact points.

If this is not true, then we can find a vertical plane that intersects the center of mass, and where all the contact points are to one side of the plane.

Since the normal forces from the surface will be upward, if they are all on one side of a plane, then the total torque about an axis through the center of mass and parallel to the plane must be non-zero. Because all the torques will have the same sign, they cannot sum to zero.

BowlOfRed
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