Question:
Two unlike parallel forces $P$ and $Q$ $(P>Q)$ act at $A$ and $B$ respectively. If P and Q are both increased by R, show that the resultant will move through a distance $d=\frac{R}{P-Q}$.
My book's attempt:
Let the resultant of two unlike parallel forces $P$ and $Q$ act at $C$.
$$\frac{P}{BC}=\frac{Q}{AC}=\frac{P-Q}{AB}$$
[Each force is proportional to the distance between the points of application of the other two.]
From 2nd and 3rd ratios,
$$AC=\frac{Q}{P-Q}AB\tag{1}$$
Let when $P$ and $Q$ are increased to $P+R$ and $Q+R$, their resultant act at $D$, where $CD=d$.
$$\frac{P+R}{BD}=\frac{Q+R}{AD}=\frac{P+R-Q-R}{AB}$$
[Each force is proportional to the distance between the points of application of the other two.]
From the 2nd and 3rd ratios,
$$AD=\frac{Q+R}{P-Q}AB\tag{2}$$
$(2)-(1)$,
$$AD-AC=\frac{Q+R-Q}{P-Q}AB$$
$$d=\frac{R}{P-Q}AB\ (\text{showed})$$
My comments:
My book assumed that the forces $P$ and $Q$ acting at $A$ and $B$ respectively will have a resultant force $P-Q$ that will act at $C$. I take issue with this. I would've agreed with the book if P and Q were both acting at the same point; then the resultant would've been $P-Q$, which would've acted at the same point as $P$ and $Q$. However, that's not the case here. P and Q are acting at two different points in opposite directions. I think what will happen is that the body that the two forces are acting on will experience a net torque and start rotating.
My question:
- Isn't my book's attempt wrong?
Best Answer
The center of mass of the board will accelerate given a non-zero net force, and the board will rotate about the center of mass given a non-zero net torque, but the problem asks to find the effective movement of the point of application of the "resultant". For this problem, the "resultant" is a simply a single force that would cause the same motion as the set of actual forces.
Sometimes a single effective force is insufficient to model the problem; for example, equal forces that cause rotation, and a "couple" can be used in this case. A couple is a system of forces whose vector sum is zero.
It can be shown that every system of forces is equivalent to a single force through an arbitrary point plus a couple (either or both of which may be zero). [Symon, Classical Mechanics]
Resolving a system of forces into single force and a couple is common in engineering mechanics textbooks, but not so much in physics mechanics textbooks.
The problem provides insufficient information to evaluate the motion. To evaluate the motion (translation and rotation) you need the forces (magnitude and points of application), the mass and length of the board, and the density of the board as a function of length. Then you can calculate the motion of the center of mass from the net force and the rotation about the center of mass from the net torque.