Kvant Physics Problem 176

Verified: no
Verdicts: SKIP + SKIP
Solve time: 1m09s
Source on kvant.digital

Problem

Published in the journal by mistake under number 166.

Why is it easier to pierce a hole with an awl if the awl is rotating? Why must a nail be rotated in order to pull it out of a wall? Why, when cutting bread or meat, do you move the knife back and forth, whereas when cutting cheese you only press on the knife?

A. G. Kosourov

Setup and Assumptions

The physical system consists of a cutting or piercing tool, such as an awl, nail, or knife, interacting with a solid material. The quantity of interest is the force required to produce relative motion between the tool and the material.

Let $N$ be the normal force exerted by the material on the tool, measured in newtons. Let $\mu$ be the coefficient of friction between the contacting surfaces, dimensionless. Let

$$F_f=\mu N$$

be the friction force opposing sliding.

The materials are treated as deformable solids. The contact region between tool and material is assumed to be small compared with the dimensions of the bodies. Air resistance is neglected. The motion is sufficiently slow that inertial effects are unimportant. The main resistance to penetration or withdrawal is assumed to arise from friction along the lateral surfaces of the tool and from the mechanical separation of the material.

Physical Principles

Dry friction is described approximately by Coulomb's law,

$$F_f=\mu N.$$

The friction force always acts opposite to the relative sliding velocity of the contacting surfaces.

Mechanical work is given by

$$dA=\mathbf F\cdot d\mathbf r.$$

For a given displacement, the required external force decreases if part of the work is supplied through motion in another direction.

Cutting occurs when stresses concentrated near a sharp edge exceed the strength of the material. Relative tangential motion of the edge with respect to the material increases the local shearing action and reduces the normal force needed to create fracture.

Derivation

Consider first an awl penetrating a material. If the awl is pushed straight inward, every point of its lateral surface tends to move only in the axial direction. The material presses on the sides of the awl with a normal force $N$, producing a friction force

$$F_f=\mu N$$

directed opposite to the penetration.

The external force required to advance the awl must overcome both the resistance associated with deforming and separating the material and the friction force along the sides.

Suppose now that the awl rotates while being pushed. The surface of the awl acquires a tangential velocity component. Relative motion between the awl and the surrounding material is no longer purely axial. The friction force acts opposite to the total relative motion and thus is no longer directed entirely against penetration. Part of the friction force acts in the circumferential direction and is balanced by the applied torque.

As a result, the axial component of the resisting friction force becomes smaller than in the nonrotating case. A portion of the required work is supplied through rotational motion. The force needed for penetration is reduced, which is why a rotating awl enters more easily.

The same reasoning applies to pulling a nail from a wall. The wall presses on the nail and creates substantial friction along its surface. If the nail is pulled straight out, the entire friction force opposes extraction. When the nail is rotated, relative sliding occurs around its circumference. Part of the friction force is then associated with rotational motion rather than purely axial motion. The axial resistance decreases, so the nail can be withdrawn with a smaller pulling force.

Consider next a knife cutting bread or meat. The cutting edge must create sufficiently large stresses to break the material. If the knife is pressed straight downward, the edge acts mainly by compression. When the knife is moved back and forth, the edge simultaneously exerts a tangential action. The material near the edge experiences strong shear deformation in addition to compression.

Fracture under combined compression and shear occurs at a smaller normal force than fracture produced by compression alone. The cutting process becomes easier because the horizontal motion contributes mechanical work directly to the separation of the material.

Cheese behaves differently. It is comparatively soft and yields readily under compression. The required stresses can be produced simply by pressing the knife downward. Additional sawing motion contributes little because the material does not require substantial shearing to initiate separation. The knife therefore cuts effectively with predominantly normal pressure.

Result

No numerical calculation is required.

The governing relation is the friction law

$$F_f=\mu N.$$

Rotation or reciprocating motion changes the direction of relative motion between the tool and the material. Consequently, the friction force is no longer directed entirely against penetration or extraction. The axial resisting component becomes smaller, and part of the required work is supplied by torque or horizontal motion.

For this reason:

1. A rotating awl penetrates more easily than a nonrotating awl.
2. A nail is easier to pull from a wall if it is rotated while being withdrawn.
3. Bread and meat are cut more effectively when the knife is moved back and forth, because the edge then produces shear as well as compression.
4. Soft cheese can usually be cut by simple pressure, since compression alone is sufficient to separate the material.

$$\boxed{\text{Rotation or sawing motion reduces the required axial cutting or pulling force by redirecting friction and adding shear deformation.}}$$

Sanity Checks

The explanation is consistent with the limiting case $\mu=0$. If there is no friction, rotation of an awl or nail provides little advantage because there is no frictional resistance to redistribute.

If friction is large, $F_f=\mu N$ becomes large, and the benefit of converting part of the interaction into rotational or tangential motion becomes more significant.

The dimensions are consistent. The coefficient $\mu$ is dimensionless, while both $F_f$ and $N$ have units of newtons.

The step most vulnerable to incorrect reasoning is the interpretation of the friction force. Rotation does not eliminate friction. It changes the direction of relative motion, causing the friction force to acquire components that are balanced by torque or horizontal motion rather than entirely by the axial pushing or pulling force.