Why is it possible to keep an object, say a book, horizontal while holding its corner?

When holding a book from its corner with two fingers in a pinched position, the fingers act as a sort of hinge, and the book is free to rotate about this corner only. The external forces acting on the book are $$mg$$, and the contact forces from the fingers. Since contact forces act on the axis of rotation, they don't exert any torque. $$mg$$ would exert a torque unless the book's center of gravity is directly above or below the "hinge".

But then, why is it possible to keep a (reasonably sized) book horizontal while holding from its corner?

Edit: Since the question is somewhat ambiguous in the orientation of the book, here is an illustration of what I was talking about (except the finger should be in a pinched position):

• Are you ignoring friction here? Commented Jan 10 at 13:43
• Now that you edited the question, I highly doubt you are able to hold the book horizontally like that IF your fingers really only touch the book at two points. If the book is light, probably part of your finger at the finite area where you touch provides the torque needed to balance gravity. I am not able to hold a heavy book like that horizontally. :-D Commented Jan 10 at 14:44
• Try this while gripping the book with a pair of needles instead of your fingers - you won't be able to keep the book level. Your fingers are not point objects that apply force only to the exact axis of rotation, but the needles are a better approximation. Commented Jan 10 at 20:52
• @brainfreeze: The currently accepted answer (Koschi) assumes the book is held parallel to the ground, while the updated question clearly shows the book is perpendicular to the ground. Please accept a different answer. Commented Jan 10 at 21:27
• The fact that you can hold a single piece of paper like this and rotate it in any way tells you that your fingers must be applying torque. For all of the weak-fingered of the world, imagine it's a children's book or a light paperback. The drawing isn't to scale, but OP never said that it had to be The CRC Handbook. Commented Jan 10 at 22:34

8 Answers

Since contact forces act on the axis of rotation, they don't exert any torque. mg would exert a torque unless the book's center of gravity is directly above or below the "hinge".

You are assuming that fingers act as point objects, which they are not. The contact area of the finger around the pinching point is finite, and you can have forces that act away from the axis of rotation. This allows the friction to create a counter torque to balance the torque due to the gravity of the book.

Key: (The orange thing is a finger, black circle is the area of contact with the book, and the red cross is the axis of rotation)

• I was going to upvote just on the words then I scrolled down and saw the masterpiece and now I'm disappointed I can only upvote it once. Commented Jan 10 at 21:43
• how does the pressure from the pinch affect the torque? Commented Jan 12 at 13:07
• @CapiEtheriel the pressure doesn't change the torque in the static case. It could change the area of contact, and would increase the total possible torque, but since the torque is from static friction, it will balance out T_mg until the static condition is no longer met.
– Sam
Commented Jan 12 at 18:15
• I recommend the "Fill" tool to fill the whole finger with orange (unless it's an artistic statement) Commented Jan 15 at 21:39

If you consider the static friction involved it is possible to achieve balance. By contrast, trying to do this with a block of ice would be almost impossible.

The net squeezing forces from the top $$M_T$$ and bottom $$N_B$$ must counter act gravity $$W$$

$$N_B - N_T = W$$

In the horizontal direction, forces must balance

$$F_B - F_T = 0$$

and net torque about the pivot must counteract the moment arm

$$\tfrac{h}{2} ( F_T + F_B ) = \tfrac{\ell}{2} W$$

subject to the friction constraints $$F_T < \mu N_T$$ and $$F_B < \mu N_B$$

Results depend on how much we squeeze with $$N_T$$

$$F_B = F_T = \frac{ \ell W}{2 h}$$

and

$$N_B = W + N_T$$

subject to the limit

$$N_T \gt \frac{\ell W}{\mu 2 h}$$

As you can see, as the coefficient of friction $$\mu$$ goes down, the requirement for the top pinch force $$N_T$$ goes up significantly in a $$1/x$$ fashion.

Do you see why for example people used to have ice pinchers to carry ice blocks? These maximize friction and pinching force compared to holding the ice with just a glove.

• Ice is a very special material, it can't really be compared to anything else. It doesn't have a proper coefficient of friction at all, because its sliding properties are strongly influenced by layer melting at the contact surface. Ice pinchers don't work by friction, they work by punching little holes into the side of the ice block and holding on to those. Commented Jan 11 at 9:55
• @leftaroundabout - and the effect of the little holes is to provide friction above what would be possible otherwise, or as I said to increase the friction force. Commented Jan 11 at 13:32
• Great, now I need to figure out how to get a rectangular block of ice... Commented Jan 11 at 20:19
• The picks/holes don't rely on friction at all, just gravity and the structural strength of the ice. The block hangs on the picks, it isn't pinched between them like the book between finger/thumb.
– Nij
Commented Jan 11 at 22:50
• @leftaroundabout Ice is a special material in the narrow region of temperature that we mostly experience it - at most temperatures, ice absolutely has a coefficient of friction - see skiing below about -35C. Commented Jan 11 at 23:17

Interesting question, I actually had to take a book to empirically check my answer.
The axis of rotation is defined by where the finger on the bottom of the book (the side nearer to the ground) touches the book. To balance the book horizontally, the finger on the top has to apply a torque acting in the opposite direction to the torque gravity applies by pulling on the center of gravity of the book.
This is usually done by grabbing the book on the top a little bit more to the edge than on the bottom and pushing down on the book with the finger on the top. This offset between finger on the bottom and finger on the top can be small, but if you try moving the finger on the top more to the center then the bottom one, you will see that you cannot hold the book horizontally.

The key to this situation is the fact that the finger does not apply a force only at the axis of rotation. It actually provides force over an area, specifically the area of the pad of your finger. This means that, for any rotation axis you draw, the force of friction for nearly any part of the pad isn't through the rotation axis so it can induce a torque that keeps the book up. It's a small distance from any one point under the finger to the axis of rotation, so the torques are small per unit force, but they are nonzero. With enough clamping force, they can indeed generate the required torque to keep the book up.

Were you to replace the fingers in the scenario with the tips of a carpentry nail, the situation would be harder. The tip of a nail is much smaller, so its induced torques are much smaller given the same forces. This means that you have to push much harder on the nail to generate enough friction forces to hold the book up from nailpoint. In fact, you may have to push so hard that you exceed the material strength of the book and end up driving the nail through the book before generating enough torque due to friction.

In the opposite scenario we can consider a bolted joint: two pieces of metal held together by bolts. Obviously such metal joints are very good at providing the required torque. Despite what your intuition may say, the strength of a bolted joint is not caused by sheering, where the edges of the metal push on the side of the bolt. Bolts are generally weaker in this direction. Instead, we use the bolt in its strong direction to clamp the pieces of metal together, exactly like you're doing when pressing on the book with your fingers. This is the strong direction for the bolts, and it allows all of the bolts to share the load.

As a result, not only is this approach sufficient to hold up a book from the corner, but it also holds up skyscrapers and bridges. If you are interested in more information on how these wonderful joints work, I highly recommend The Incredible Strength of Bolted Joints, an incredibly accessible video by The Efficient Engineer.

In essence there are three forces acting on a book as shown below.

There are various ways of analysing the situation but one way is to regard the thumb as the pivot and then you have a balance "seesaw".
Overall the net forces is zero and the net torque is zero, so there is static equilibrium.

PS Doing the same with fingers was a little difficult for me as I needed to take the photograph.

Why is it possible to keep a (reasonably sized) book horizontal while holding from its corner?

Friction. The key point is to realise that you are typically holding the book between a finger and a thumb, and your thumb in particular is in contact with the book over a significant area, not at a single point. Friction between your thumb and the book acts at a small distance away from the pivot and so exerts a torque on the book.

Two observations which show that friction is involved are (a) this is more difficult to achieve on a book with a smooth cover than a book with a rough cover; and (b) it is also more difficult to achieve if you reduce the area of contact e.g. try to hold the book with the pressure from two fingers on opposite sides instead of a finger and a thumb.

(I think the answers which rely on pivoting have misunderstood the orientation of the book. My understanding is the book is held by pressure on its two sides, not top and bottom.)

I could hold and balance a heavy book horizontally by thumb force F1 pressing down at corner (red, origin O) and two vertical upward reactions F2,F3 (green) through my left hand index finger knuckle high points entirely balancing the book weight W (red).

Static equilibrium of forces and moments (without considering dynamics) is adequate to find three forces (F1,F2,F3) in terms of W and finger phalange distances (a,b) by solving three statics equations:

Taking moments about x-axis $$a( F2+F3)= WL$$ Taking moments about y-axis $$F3~ b =WB$$ Force balance z-axis $$( F2+F3)= F1+W$$

Keeping an object, like a book, horizontal while holding its corner is possible due to the principles of physics related to torque and balance. Torque is the tendency of a force to rotate an object around an axis, and balance is achieved when the torques on an object are equal and in opposite directions.

In the case of holding a book by its corner, the force of gravity acts downward on the book's center of mass. When you hold the book at the corner, you apply an upward force with your hand. The book will remain horizontal when the torque created by the force applied by your hand is equal to the torque created by the force of gravity acting on the book's center of mass.

If you imagine the book as a uniform object, its center of mass is typically at its geometric center. When you hold the book at one corner, the distance from your hand to the center of mass creates a torque. The force of gravity acting on the center of mass also creates a torque. For the book to remain horizontal, these torques must balance each other out.

Essentially, by adjusting the angle and position of your hand, you are applying a force that counteracts the torque produced by gravity. Achieving this balance allows you to keep the book horizontal while holding its corner. It's a dynamic equilibrium where the torques and forces are in harmony, resulting in a stable and horizontal position for the object.