Week 2: Newton’s Laws:
Continued
Summary
We now continue our discussion of dynamics and Newton’s Laws, adding a few more very important force rules to our repertoire. So far our idealizations have carefully excluded forces that bring things to rest as they move, forces that always seem to act to slow things down unless we constantly push on them. The dissipative forces are, of course, ubiquitous and we cannot a ord to ignore them for long. We’d also like to return to the issue of inertial reference frames and briefly discuss the topic of pseudoforces introduced in the “weight in an elevator” example above. Naturally, we will also see many examples of the use of these ideas, and will have to do even more problems for homework to make them intelligible.
The ideas we will cover include:
•Static Friction 49 is the force exerted by one surface on another that acts parallel to the surfaces to prevent the two surfaces from sliding.
a)Static friction is as large as it needs to be to prevent any sliding motion, up to a maximum value, at which point the surfaces begin to slide.
b)The maximum force static friction can exert is proportional to both the pressure between the surfaces and the area in contact. This makes it proportional to the product of the pressure and the area, which equals the normal force. We write this as:
fs ≤ fsmax = µsN |
(138) |
where µs is the coe cient of static friction, a dimensionless constant characteristic of the two surfaces in contact, and N is the normal force.
c)The direction of static friction is parallel to the surfaces in contact and opposes the component of the di erence between the total force acting on the object in in that plane – it therefore acts to hold the object stationary until the applied force exceeds the maximum fsmax. Note that in general it does not matter which direction the applied force points in the plane of contact – static friction usually acts symmetrically to the right or left, backwards or forwards and required to hold an object stationary.
•Kinetic Friction is the force exerted by one surface on another that is sliding across it. It, also, acts parallel to the surfaces and opposes the direction of relative motion of the two surfaces. That is:
49Wikipedia: http://www.wikipedia.org/wiki/Friction. This article describes some aspects of friction in more detail than my brief introduction below. The standard model of friction I present is at best an approximate, idealized one. Wikipedia: http://www.wikipedia.org/wiki/Tribology describes the science of friction and lubrication in more detail.
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96 |
Week 2: Newton’s Laws: Continued |
a)The force of kinetic friction is proportional to both the pressure between the surfaces and the area in contact. This makes it proportional to the product of the pressure and the area, which equals the normal force. Thus again
fk = µkN |
(139) |
where µk is the coe cient of kinetic friction, a dimensionless constant characteristic of the two surfaces in contact, and N is the normal force.
Note well that kinetic friction equals µkN in magnitude, where static friction is whatever it needs to be to hold the surfaces static up to a maximum of µsN . This is often a point of confusion for students when they first start to solve problems.
b)The direction of kinetic friction is parallel to the surfaces in contact and opposes the relative direction of the sliding surfaces. That is, if the bottom surface has a velocity (in any frame) of ~vb and the top frame has a velocity of ~vt 6= ~vb, the direction of kinetic friction on the top object is the same as the direction of the vector −(~vt −~vb) = ~vb −~vt. The bottom surface “drags” the top one in the (relative) direction it slides, as it were (and vice versa).
Note well that often the circumstances where you will solve problems involving kinetic friction will involve a stationary lower surface, e.g. the ground, a fixed inclined plane, a roadway – all cases where kinetic friction simply opposes the direction of motion of the upper object – but you will be given enough problems where the lower surface is moving and “dragging” the upper one that you should be able to learn to manage them as well.
•Drag Force50 is the “frictional” force exerted by a fluid (liquid or gas) on an object that moves through it. Like kinetic friction, it always opposes the direction of relative motion of the object and the medium: “drag force” equally well describes the force exerted on a car by the still air it moves through and the force exerted on a stationary car in a wind tunnel.
Drag is an extremely complicated force. It depends on a vast array of things including but not limited to:
–The size of the object.
–The shape of the object.
–The relative velocity of the object through the fluid.
–The state of the fluid (e.g. its internal turbulence).
–The density of the fluid.
–The viscosity of the fluid (we will learn what this is later).
–The properties and chemistry of the surface of the object (smooth versus rough, strong or weak chemical interaction with the fluid at the molecular level).
–The orientation of the object as it moves through the fluid, which may be fixed in time or varying in time (as e.g. an object tumbles).
The long and the short of this is that actually computing drag forces on actual objects moving through actual fluids is a serious job of work for fluid engineers and physicists. To obtain mastery in this, one must first study for years, although then one can make a lot of money (and have a lot of fun, I think) working on cars, jets, turbine blades, boats, and many other things that involve the utilization or minimization of drag forces in important parts of our society.
To simplify drag forces to where we learn to understand in general how they work, we will use following idealizations:
50Wikipedia: http://www.wikipedia.org/wiki/Drag (physics). This article explains a lot of the things we skim over below, at least in the various links you can follow if you are particularly interested.