Equilibrium when torque about the COM is zero?

Question

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.

Answer 1

Hope you are doing well.

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.

Lets bisect everything one by one. First they say that if a body is placed on a horizontal surface, the torque of the contact forces about the COM should be zero to maintain equilibrium.

This is true because we know that for a body to be in rotational equilibrium, sum of all the torques must be zero, but why zero about the centre of mass? Because if the object is not pivoted anywhere and if the forces are balanced(which is in the case it is kept on a surface) then the COM will not accelerate, that means that the axis of rotation must also pass through the COM otherwise if the axis of rotation does not pass through it then the COM must also move in a circle, meaning it will also accelerate which should not happen if there is no net force.

So there should be no torque along the axis of rotation which is through COM.

Second they say 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. Seems confusing?

By this they mean that a vertical line, imagine one, passing through the COM, cuts the base area, at a point where it is in contact. Why should that point of the base where vertical line from COM passes, be in contact?

Lets imagine a man of say 60kg who is holding a ball of say 120 kg in his one hand. So the COM shifts towards the ball. If you are doing this chapter then you must have done COM, and from that you know that the force of mg always passes down in a vertical line through the COM(man+ball), so there must be a normal force from the ground passing up through the leg. If the man is bent to the other side then the COM will not shift that much to the ball and so the normal force of the legs would be able to pass through the COM directly, thus balancing all the forces and also not giving it torque. If the man is not bent then the normal force of the ground will not directly pass vertically up through the COM(man+ball) but at some side, which will give some angular acceleration to them(man+ball), so they will fall. So the vertical line from the COM(man+ball) must pass through the base surface at a point within contact area(of legs).

In case of a hollow hemisphere placed with it diameter on a horizontal surface, the vertical line passing through COM still passes through the area between the contact points(which form a circle) so it is stable both in rotation and translation.

For a block on the edge of a table(with friction), we can use the principle too. If we start sliding the block with our hands, slightly towards the edge, the block will not fall unless the vertical line passing through the COM does not pass through any contact point(or area between them) with the table. So that's why you must have noticed for evenly mass distributed blocks if you make them slide through an edge at some height, they will not fall unless half of the volume has already gone to the side of the edge.

But if there is a block of mass between two tables, with say, approximate 48% contact area on right table, and 48% contact area on left table, and no contact area for the 4% centre part, the block is still stable. Why? Simply because the vertical lines passing through the COM though does not directly pass through a contact point, but is still between the area of contact points of the two sides, if you join all the points with geometry.

Answer 2

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.

Answer 3

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.