# Force Diagram on an Overhanging Book [closed]

I am a beginning physics student.

I have a question (probably very simple for you all haha) about forces on an overhanging book, off a table. We all know that the max you can push it would be half the length of the book. The pivot of the book would also be the edge of the table. If the book were to be pushed to it's max, what forces would be enacted on the book? What forces would prevent it from falling/swiveling about the pivot? It would be best if you could provide a diagram, but an explanation is fine.

Thanks!

I don't know why people down-voted your question; it’s good to clear up confusion, no matter how small. In fact, this question got me thinking for a while.

Anyway, think about all the forces on the book. You have gravity, which "acts on" the center of mass. Then, you have a normal force of equal magnitude but opposite direction, from the table onto the book. Since the normal force is equal and opposite to force of gravity, there is no net force in either direction, meaning the book will remain stationary.

Before talking about rotation, you must first understand that for the book to rotate, there must be a net torque applied. Simply speaking, torque is a force applied at a distance from a pivot point.

In the 2nd case, when the book's center of mass is pushed past the edge, there is a net torque, as gravity is acting downward from a distance x from the pivot point (imagine a clockwise rotation). At the instant after the book is released, it will begin to rotate around the pivot point, so the only points of contact lie at the pivot point, thus the contact force is acting at the pivot point. And this creates no torque, because it is a force applied at zero distance from the pivot point. Therefore, there is a net torque applied by gravity, thus it will begin to rotate.

So, as long as the center of mass is not past the edge of the table, there will be no net torque (gravity and normal force will cancel each other out), and the book will remain stationary.

Did this help? If the book were to be pushed to it's max, what forces would be enacted on the book? What forces would prevent it from falling/swiveling about the pivot?

Ignoring friction, there would be two forces acting on the book: gravity and normal reaction of the table and the balance of these forces and torques due to these forces would prevent the book from falling over the edge of the table.

The diagrams below show how the distribution and balance of these forces are changing, as the book is being moved over the edge of the table. A. The book is away from the edge. The normal reaction force, $N$, is equal to the force of gravity, $mg$.

B. The book is partially hanging over the edge. The normal reaction force is still equal to the force of gravity, but now we have to consider the torques trying to rotate the book over the edge. Since, $m_2g<m_1g$ and $d_2<d_1$, the book does not fall. The difference between the torques due to $m_1g$ and $m_2g$ is balanced out by the torque due to the normal reaction force:

$N \times d+m_2g \times d_2 = m_1g \times d_1$

Since the normal reaction force is greater than $m_1g$, it has to be applied closer to the edge than $m_1g$ - otherwise its torque alone would overpower the torque due to $m_1g$. Due to this shift, the normal reaction force has to be redistributed along the length of the book remaining on the table, as shown by the dashed green line: the normal reaction will be increasing toward the edge.

C. This is just a progression of the movement, with the normal reaction force moving even closer to the edge.

D. A little more than half of the book is overhanging, therefore the torque due to $m_2g$ is exceeding the torque due to $m_1g$ and the book is starting to rotate over the edge. The normal reaction force has shifted all the way to the edge and does not contribute to the torque.