Is a 'resultant' a single force acting on an object?

Question

If I have two forces $A$ and $B$ acting on an object, in a lot of cases (especially Archimedes law) we take these two forces and mathematically sum them, which gives the value of a single force equivalent to both acting on a point particle, or on an object around the point in which the forces act. Why do we treat this as being 'one' force?

In the case of buoyancy we take the two pressure forces and sum them, defining a new force $F_b$ the force of upthrust. Is this an acceptable thing to do? It may be confusing in that we act as if there is one force acting on the object but really this is due to the difference of pressures, should I call $F_b$ a 'single' force or the resultant of the two pressure forces?

When we have any 'force' on an object could it in fact be the resultant of several other forces acting on it?

Answer 1

When you only care about finding out $\vec{r}(t)$ (you treat the object as a point, or you simply don't care about it's orientation in space). It's suffice to vectorially add all the forces and the physics would be the same as if that vector sum of the forces is acting on the center of mass (COM) of the body (where $\vec{r}$ is the poition vector of the COM). But the moment you're interesting in the orientation of the body and how is that orientation changing with time, you need to consider the torque due to each force acting on different parts of the body.

Basically if you only want to know how the position of the body is changing with time (position of its COM to be precise), just add all the forces up to make one force and use Newton's second law. This will only give you information about body's motion in three degrees of freedom (x,y and z). But to define a rigid body in space you need six degrees of freedom: 3 to define the position of it's COM (x,y,z) and 3 to define it's orientation in space (Euler angles: $\alpha ,\beta,\gamma$). To be able to extract information about $\alpha ,\beta,\gamma$; i.e, the information related to body's orientation and it's change in time, you need to treat each force individually.

Answer 2

in a lot of cases (especially Archimedes law) we take these two forces and mathematically sum them, which gives the value of a single force

No. We are not trying to extrapolate it with a single force, but to calculate a net force. Net force is a vector sum of all acting forces on body center of mass. Then Why we use net force ? It's because second Newton law works on a net force, namely : $$ \sum_i \vec F_i = m\vec a $$

Then again, you can ask,- why 2-nd Newton law works exactly on a sum of forces ? Because usually, forces are in equilibrium and can cancel each other. Take for example a trivial case,- book laying on your table. For it applies, $$ \vec N + \vec W = m \frac {d\vec v}{dt},$$

where $\vec N$ is table normal force, and $\vec W$ - book weight. These forces are in equilibrium- they are of equal magnitude, but with opposite directions, so their sum is zero, thus we get :

$$ m \frac {d\vec v}{dt} = 0 ,$$

This means that for book, $\vec v = \text {const},$ That is it either stands still or moves in the same velocity. Which is what first Newton law says,- if object affecting net force is zero,- it maintains it's inertial properties, speed, direction, etc., until some, or better to say net force, pushes it out of inertial state.

Then again, you can ask,- Why forces can cancel each other out ? This time, I'll say - I don't know. And rhetorically will ask you in my turn,- Why what is left in your pocket is a sum of your income and expenses ?