How Far Can the Bottom Book Overhang the Table Without Falling?

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In summary, the problem involves three books stacked over the edge of a table, with the top book overhanging the middle one by 0.5L and the middle book overhanging the bottom one by 0.25L. The goal is to determine the maximum overhang for the bottom book without the books falling. Using the equations for torque and Newton's second law, the solution involves balancing the forces and torques on each book, with the sum of their moments needing to equal zero. By setting up and solving the equations for each book, the answer is found to be 0.25L.
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a_narain
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Homework Statement


Three identical books of length L are stacked over the edge of a table as shown in the figure. The top book overhangs the middle one by .5L, so it just barely avoids falling. The middle book overhangs the bottom one by .25L. How far can the bottom book overhang the table without the books falling?

RW-12-56.jpg

Homework Equations


Torque = rFsin(angle)
Newton's second law

The Attempt at a Solution


In approaching this problem, I look at the top two books first. I noticed that the force on book 3 from 2 has to equal the gravitational force of book 3, and the torque is zero using the center of gravity of book 3 as the pivot point.
A similar situation occurred for book 2, where the force of book 3 acts on the end, and while the force of book 1 and gravity act at the center of mass. From this, I got that the force of book 1 on 2 = 2mg.

My work for book 1:
-2mgx-(.5L-x)mg+(.5L-x)3mg = 0
The answer I get when I solve for x is 1/4 L, but that is wrong?

What exactly am I doing wrong, am I setting up the problem correctly?
 
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  • #2
Maybe try applying the ∑ T = 0 about the edge of the table?

Figure the distance that the CofM of each book is from that point and then the sum of their moments will need to sum to 0 right?
 
  • #3
What do you mean by the sum of their moments?
 
  • #4
a_narain said:
What do you mean by the sum of their moments?

http://en.wikipedia.org/wiki/Torque

That would be Torque, insofar as you are taking the moment about a point as in r X F.
 

1) What is static equilibrium?

Static equilibrium is a state in which an object is at rest and the forces acting upon it are balanced. This means that there is no acceleration or movement in any direction.

2) How do you determine if an object is in static equilibrium?

An object is in static equilibrium if the sum of all external forces acting on it is equal to zero, and the sum of all torques (rotational forces) acting on it is also equal to zero. If these conditions are met, the object is at rest and in static equilibrium.

3) What are the conditions for static equilibrium?

The conditions for static equilibrium include: the sum of all external forces acting on the object must be equal to zero, and the sum of all torques acting on the object must also be equal to zero. In addition, the object must remain at rest and cannot have any rotational movement.

4) What is the difference between static and dynamic equilibrium?

Static equilibrium refers to a state in which an object is at rest and the forces acting upon it are balanced. Dynamic equilibrium, on the other hand, refers to a state in which an object is moving at a constant velocity and the forces acting upon it are balanced. In dynamic equilibrium, there is still movement, but it is constant and there is no acceleration.

5) How is static equilibrium important in engineering and physics?

Static equilibrium is important in engineering and physics because it allows us to analyze and understand the forces acting on objects at rest. This is crucial in designing structures and machines to ensure they are stable and will not collapse or fail under different forces. Static equilibrium is also important in physics as it helps us understand the principles of balance and stability in our physical world.

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