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The Physics of Pegs

Posted by admin on September 3, 2011

This is Part II of my three-part series on how a violin peg works.  This section covers the physical forces at work on the peg.

The Physics

There are actually three forces at work against the peg: the normal force of the pegbox, the frictional force of the pegbox, and the tensional force of the string.  Ideally, the normal and frictional forces counteract the tensional force.

Normal Force

If you are like me, you probably thought I meant "regular" or "usual" force.  But no, normal force is a very specific type of force, defined as the repulsive force of interaction between atoms at close contact.  Maybe you've heard of Pauli interactions at a quantum level?  If not, just consider the force that keeps an object on top of a table, instead of dropping straight through to the ground.

Normal force is exerted at right angles to the surface in question.  Therefore, the strength of the normal force depends on the steepness of the taper.  Violins and violas use a taper of 1:30, meaning that for every 30 mm of distance, the shaft changes diameter by 1 mm.  Cellos have a steeper taper of 1:25.  Pre-20th century instruments often have what is now considered an excessively steep taper of 1:20.  The steeper the taper, the greater the normal force being exerted. 

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Frictional Force

Friction is a surface force that opposes relative motion, and is derived from the normal force.  In the words of my engineering father:

"The normal force generated as the peg is shoved into the tapered hole creates a frictional force which equals the normal force X coefficient of friction.  Since the peg is moving during installation the frictional coefficient used is the kinetic coefficient but once the peg is in position and no longer moving the friction becomes based on the static coefficient of friction.  The peg stays in place because the static coefficient is greater than the kinectic coefficient so it takes more force to remove the peg than it took to insert the peg."*

To explain: Each type of material has two coefficients of friction- while moving (kinetic) and while at rest (static).  To calculate the coefficients of friction for pegs, we'd need to take into account the material of the peg, the maple pegbox, and any peg dope that has been applied to the peg.  The peg stays in place because of static friction.  To turn the peg, you have to exert enough force to overcome the static friction, but after that, less force is needed to counteract the kinetic friction.  Just think about how you have to turn a bit harder to get the peg started, then ease off the pressure as you bring the string up to pitch. 

Tensional Force

Tension is the magnitude of the pulling force exerted by a string on another object (the peg). There is a whole library of fascinating literature about string tension, and someday I hope to be able to understand it all properly.  Suffice it to say that in a properly working violin peg, the normal and frictional forces must be greater than the tensional pull of the string.  Remember that each peg is only connected to one string, so we are looking at this on an individual basis; many of the articles talk in terms of total string tension.

Synopsis

The peg system consists of three variables which must be properly balanced to work: the steepness of the taper (normal force), the peg material (frictional force), and the tautness of the string (tensional force).  As the conscienscious violinmaker will realize, changing any of these variables will effect another part of the violin outside the peg system.  For instance, string tension can be changed, but at the sacrifice of pitch or tone.  Peg material may be changed, but could influence the durability of the pegs.  And the steepness of the taper could be increased for better sticking power- if you were paying attention during the physics lesson, this will increase both normal force and by extension, frictional force.  But, as you will see, changing the taper could create more problems down the road.

For more fun with forces, check out Fizzics Fizzle.

*Understanding how normal force relates to frictional force, English version:

  1. Imagine trying to move a refrigerator across an ice rink.  It's not too hard because there isn't much friction.
  2. Imagine trying to move a refrigerator across your kitchen.  The friction of the floor makes it harder.
  3. Imagine trying to move a refrigerator across your kitchen while an elephant sits on top.  That's really hard because the elephant adds a lot of normal force, which increases the friction.
  4. Imagine trying to move a refrigerator across your kitchen while an elephant sits on top and a mouse creeps across the room.  The elephant and mouse nullify each other, so we're back to #2.
Special Thanks to the Crawford Consulting team:
Steven Crawford, Chief Engineer
Susan Crawford, Copy Editor
Brian and David Crawford, Junior Engineers

In the final section on pegs, I will cover peg problems and solutions.

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Still confused?  Leave a question and I will post an answer.

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