Motions of the the two bodies in a direction
parallel to the touching surfaces is prevented (or hampered) due to
molecular adhesion and/or irregularities on the involved surfaces
often too minute to discern with the nacked eye.
Dry Friction occurs when the surfaces in contact with each other
are free of any lubricants.
Motions of the the two bodies in a direction
parallel to the touching surfaces is prevented (or hampered) due to
molecular adhesion and/or irregularities on the involved surfaces
often too minute to discern with the nacked eye.
In this course we are dealing only with dry friction.
Wet friction occurs when the surfaces of two solid bodies
are not directly in contact but separated by a thin film of lubricants.
Again friction will try to hamper the motion but the underlying
physics is related to fluid mechanics which is not part of this course.
Above distinction is that of two extreme case with mixed friction inbetween which often occurs during start-up of motion when the separating lubricant film is not fully developed.
| In the figure 8.2a a block of weight W is shown resting on a surface and subject to a force P. Assume both, P and W are given. If the block is not moving then the surface underneath has to exert onto the block a force which exactly counteracts the forces P and W. It is best to represent this force by two forces : force N which is perpendicular and force F which is parallel to the surface the block rests upon. |
Force N, often called the normal force (normal=perpendicular), is directed perpendicular to the direction of possible motion (of the block) and can be thought of as the resistance of the surface against penetration by the block. The resistance of course is spread in some fashion over the entire contact area and hence N is a replacement force for this resistance.
The force F is called the friction force. It is parallel to the direction of possible motion and acting onto the block in the direction opposing that of possible motion (the motion the block would undertake if friction were absent).
|
If the block is at rest the equilibrium equations are satisfied. |
In the example of Fig.8.2a that means :
) ) irrespective of how the block manages to stay put (glue,friction,nails etc.).
If friction is holding the block in place we know from experience that if the horizontal component of force P is getting too strong the block will ultimately start to slide. The critical case at which sliding just is about to set in is referred to as the case of impending motion and the value of the friction force in that case is called the maximum friction force, Fm.
Many experiments have been conducted to find out as to what influences the value of Fm. Luckily, nature is kind to us in the case of dry friction. As it turns out Fm does only depend on the normal force N and the type of materials involved and the surface roughnesses. Furthermore, the relationship between Fm and N is linear for given surfaces :
(8.2a)
Fm =  N
|
Equat. (8.2a) is referred to as the law of dry friction and
as the coefficient
of static friction. Its value is listed below for a
variety of combinations of materials the block and supporting surface
are made out of.
| Coefficients of Static Friction | ||||||||||||||||||||
|
Here is a fairly simple experiment to measure the coefficient of static friction.
|
A block is placed on an inclined plate and the angle of inclination,
 , is varied
carefully from zero to that at which sliding of the block is impending.
This angle is referred to as the angle of (static) friction.
|
The analysis of the Free Body Digram of Fig. 8.2b (sum of the force along the incline and sum of the force perpendicular to that) gives the relationship :
|
(8.2b)
=
tan()
|
| Although this course is dedicated to statics for reasons of completeness let's consider the case of motion of the block at constant velocity. Again the equilibirum equations hold (at constant velocity the accelearation is zero, right ?) and the horizontal component of the pushing force P is counteracted by the friction force F. |
(8.3a)
Fkin =  N
|
where is the
coefficient of kinematic friction. Again the value of this coefficient
depends on the involved materials and surface properties. At least for
moderate velocities it is independent of velocity.
For all materials I know of we observe :
|
(8.3b)
<
|
which reflects our experience when pushing a piece of furniture from here to there. The hardest part is to get it moving. And the best strategy is to keep it moving until it arrives at its final resting place.
|
In this section we have a device ( a horizontal lever in the figure)
which experiences friction with another item ( the vertical
rod in the figure ) at two distinct points, A and B.
Possible questions in such problems revolve around whether the device (the lever) will move under influence of external forces (the load P) or whether friction will hold it in place. Often one needs to investigate what are minimum or maximum values of design parameters of your device ( the height h or the distance L for example ) which either make motion possible or prevent it. |
|
As usual you will have to draw the relevant Free Body Diagram and establish the equilibrium equation and invoke the law of dry friction at each of the contact points.
| In the figure to the left we show a crate with a weight of 100 lbs resting on two legs A and B which are 1m apart. A force P is intended to push the crate sidewise but several questions arise. If P is applied very low ( very small h ) the crate actually might start to slide once P reaches a critical value. For larger values of h sliding though might never set in : rather an applied P will cause the crate to tilt over to the right. |
As usual our analysis starts with the Free Body Diagram
Figure 8.7a Top and Side View of screw through a plate |
Forces of thread of plate onto thread of screw
 Figure 8.7b Screw moving in direction of load P 
Figure 8.7c Screw moving against load P |
Figure 8.7a displays the top and side view of a screw going through a plate
of which only a small section is shown. The screw is subject to an axial
load P [N] and a torque M [N m] which is turninthe screw.
Friction of course
tries to prevent the turning. Subject of this section is to determine
the torque M in terms of the load P, the geometry of the
screw ( mean radius r and lead L as well as the coefficent
of dry friction, 
, between
the threads of the screw and the plate material.
If in Figure 8.7a a moment M is acting in clockwise direction,
the shown screw will move
downward in the same direction load P is pointing, if the moment is
acting counter-clockwise the screw will be moving upwards against the
direction of the external load P. In either case, the desired
relationship for the moment M needed to achieve this can be determined
from equilibrium considerations, specifically from looking at the
sum of the forces along the axis and the sum of the moments
around the axis of the screw.
Although the action of the thread in the plate onto the thread
of the screw is distributed more or less evenly across the entire contact
area we can represent this by a single normal force , N,
acting perpendicular and a single friction force, F, acting parallel
to the inclined surface of the thread.
|
Figure 8.8a shows a typical belt-drive with both pulleys rotating in the
clockwise direction. One of the pulleys is usually connected to
some motor and is referred to as the driving pulley, the other pulley is
then the driven pulley.
For the sake of discussion let's assume that pulley A is driving
and rotating in the clockwise direction as shown. Hence the torque the
motor is exerting onto this pulley is in the clockwise direction as well. |
| As a result, the tension T2 in the right section of the belt is higher than T1 in the left section in order to rotate pulley B in clockwise direction against the resistance of the machinery hooked up to this pulley. In Fig. 8.8b the moment MB, representing the action of the driven machinery onto the axle of the pulley, is therefore acting in counter-clockwise direction. |
|
| (8.8a) (T2 - T1 ) rB = MB |
(8.8b)
(T2 + T1 ) cos( ) + Py = 0
|
|
(8.8c)
(T2 - T1 ) sin() + Px = 0
|
= 90 - 
/2
Px,Py = support force of
bearings onto axle of pulley B.
|
, 
= angular
velocities
As far as friction is concerned, the only case of interest
is that of impending slippage between belt and pulley.
In that case an additional relationship between the tensions T1
and T2 can be established.
| This relationship is derived from studying the equilibrium of the belt, here by considering pulley B (pulley A can be looked at as well with identical results). On a global basis the piece of belt running over the pulley B is subject to the tensions T1 and T2 and the pulley pressing outward against the belt on every sqare inch where pulley and belt are in contact. Accompanying this normal pressure is the friction with which the belt "drags" along the pulley in clockwise direction. The friction force the belt experiences is than of course in counter-clockwise direction. |
|
In Fig. 8.8d we look at a small segment of the belt extending from
 to +
 . The angle
would be zero if the segment would start at the most counter-clockwise point
at which belt and pulley are in contact. If the belt segment is located at
the most clockwise point of contact we would have :
+ =
= angle of contact
N = normal force from pulley onto belt segment
Note that between either of the two tensions and the friction force F
we have the angle |
For the x-y coordinate system as indicated in Fig. 8.8d we can write down now the sum of the forces equal to zero in the x- and for the y-direction, respectively.
Solving the above 2 equations for forces F and N and substituing into the
law of friction and letting the angle
become very small we obtain
a differential equation. This can be solved to finally give us :
|
The contact angle has to
expressed in radians.
Equations (8.8a) and (8.8e) are usually used to determine the tensile force
in each belt section necessary to avoid slippage of the belt on the
smaller of the two pulleys for given torques.
Equations (8.8d) relates the torques and angular velocities of the two
pulleys to each other, while Equation (8.8b) is being used to determine
the support forces needed to hold the axis of a pulley in place.
= tension at right end of segment
