Coat of arms of Isaac Newton (1643-1727) When two dry solids at rest press against each other they won't move as long as the force applied tangentially to their common surface of contact does not exceed a certain limit, which is roughly proportional to the normal force that presses them against each other.
The coefficient of proportionality is called the static coefficient of friction and is often denoted by the symbol m or ms (the Greek letter mu is pronounced mew ).
As soon as such solids start moving, a lesser friction force appears in the direction which opposes the motion. It is also proportional to the normal force applied but the coefficient of proportionality (called kinetic coefficient of friction or sliding coefficient of friction) is always less than in the static case described above. This means that, unless the normal force changes, a constant tangential force large enough to overcome static friction will impart motion at a positive (nonzero) acceleration.
By definition, the "tangential force" is in the same plane as the surface of contact and the "normal force" is perpendicular to it.
The above is a valid approximation for dry surfaces under moderate pressure. At high pressure, seizure can occur, whereby the two solids will not slide at all under any reasonable force. This happens, in particular, for polished surfaces between solids consisting of the same material: Pressure makes the interface between the two solids effectively disappear and they may well behave as a single crystal, with large binding forces between nearby atoms. The solids may well break before they slide.
When a liquid lubricant is used, the above does not apply at all. Instead, the slightest tangential force will produce sliding and the resistive forces depend on the speed of sliding (the normal forces are then largely irrelevant).
Friction
Coefficients |
Table
of coefficients of friction |
Engineer's Edge
Friction
and
Tribology
by Roy Beardmore.
Let's use the following notations:
Because of Newton's second law, the following 3 equations hold:
Therefore:
T = m3 (g - b) and F = m3 (g - b) - m1 b = m2 a
Under the static regime where m1 and m2 are at rest with respect to each other, we have a = b so that the last of the above equations boils down to:
m3 g = (m1+m2+m3) a Therefore:
F = m2 a = g m2 m3 / (m1+m2+m3)
The coefficient of static friction must exceed the ratio of F to the normal force (m1 g). Thus, in the static regime, we must have:
F / (m1 g) = m2 m3 / [m1 (m1+m2+m3)] < 0.539
With the given values of the masses, that inequality doesn't hold (since the left-hand side is more than 0.711). We are thus faced with the kinetic regime where a and b need not be equal. Instead, motion is determined by the following additional equality, involving the kinetic coefficient (u).
F = u m1 g [where u = 0.439]
Therefore, the acceleration of m2 is:
a = F/m2 = g u m1/m2 = 0.282 g = 2.767 m/s2
The acceleration b of m1 (and also of m3) is obtained as follows:
F = g u m1 = m3 (g-b) - m1 b
Therefore:
b = g (m3 - u m1) / (m1 + m3) =
0.544 g = 5.334 m/s2
Finally, the tension of the string is:
T = m3 (g - b) = g (1+u) m1 m3 / (m1+m3) = 35.4 N
Let m be the static coefficient of friction between the floor and the ladder (whose other end is resting against a frictionless wall, consisting of something very slippery like wet ice or glass).
Let L be the length of the ladder and q be its inclination (the angle between the ladder and the horizontal). Let's introduce a parameter k that indicates the position of the center of gravity of the loaded ladder. The combined center of gravity of the ladder and the person who climbs it (if any) is located in the vertical line which intersects the ladder at a distance k L from its bottom (for an ordinary ladder with nobody on it, k = 0.5).
At equilibrium, the ladder does not move vertically or horizontally, so the sum of the horizontal components of all existing forces is zero; so is the sum of all vertical components.
Furthermore, the ladder doesn't rotate. Therefore, the torques of all applied forces cancel. The torques about the point of contact with the floor reduce to the torque of the vertical weight (of magnitude Mg) and the purely horizontal force (F) exerted by the frictionless wall. Therefore:
(k L cos q) Mg = (L sin q) F
The ladder will not move as long as the tangential friction force (of magnitude F) exerted by the floor does not exceed its allowed maximum for the applied normal force (of magnitude Mg).
F < m Mg
Combining the two relations above, we obtain:
k Mg / tan q < m Mg
Which is to say that q > arctg ( k / m )
The unloaded ladder will stay in place as long as this inequality is satisfied for k = 0.5 but it's safe to climb to the last rung only when the equality is satisfied when k is nearly equal to 1.
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Rolling Friction
by Ron Kurtus.
Rolling
Resistance (Wikipedia) |
Railway
Industry Overview (© 2003, AREMA® )
(The pseudo-periodicity of earthquakes.)
MIT 8.01 Some of Walter Lewin's best lines (Fall 1999)
When a ball dropped from a height h bounces off the floor to a height r h, the dimensionless coefficient r is called the coefficient of restitution.
Technically, the coefficient of restitution is defined as the ratio of the initial to final closing speed (with a change of sign that makes e positive in the common case where the closing reverses direction after a shock). It's thus the rato of the speed of approach to the speed of separation.
The above values correspond to the so-called normal coefficients of restitution, which are appropriate for head-on collision with little or no tangential speed. Otherwise, if the relative velocity of the two objects is not perpendicular to their surface of contact, the ratio of the final to the initial tangential speed is defined to be another type of coefficient of restitution whose value is only very loosely related to the normal coefficient of restitution discussed so far.
Resilience is the ability to bounce back.
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The coefficient of restitution for collisions of
happy balls, unhappy balls, and tennis balls by Rod Cross.
Coefficients of Restitution
by Jamin Bennett & Ruwan Meepagala (Physics Factbook by
Glenn Elert)
Thread:
Coefficient of Restitution
by ajinx999 (Physics Forums, 2008年06月30日)
Video:
Bouncing Balls
by Roger Bowley
(University of Nottingham) in
Sixty Symbols by
Brady Haran.
Happy and Sad Balls:
Neoprene (happy) and polynorbornene (sad).
[ Evil Mad Scientist ]
Leornardo da Vinci's Bridge (5:02)
by Mark Deegam (2016年07月16日).
Dowel routing for the da Vinci bridge puzzle (6:08)
by Pocket83 (2016年08月19日).
