Further Applications of Newton s Laws  Friction Static and Kinetic Friction


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1 urther pplications of Newton s Laws  riction Static and Kinetic riction The normal force is related to friction. When two surfaces slid over one another, they experience a force do to microscopic contact points connecting and breaking. There are two situations: Static friction where the two surfaces are not moving relative to one another Kinetic friction where the two surfaces are in motion with respect to each other. The static frictional force on an object is given by: f max s = µ s N where µ s is called the coefficient of static friction, fs max is the maximum static friction and N is the normal force. The frictional force can be smaller than this if the object is in equilibrium (not moving) up to this maximum value. The kinetic frictional force on an object is given by: f k = µ k N The kinetic friction is assumed to be constant regardless of the motion. The coefficients of friction for the static and kinetic situation are different. In general, the static coefficient is larger than the kinetic coefficient. See page 167 for a table of µ s and µ k for common surfaces. Note that friction always opposes the motion. riction is just another force that can be considered when we do Newton 2 nd problems. However, we have to be careful and make sure the sign is correct direction. Let s do problems 54, 56, 512, 517 and 518 as examples
2 Drag orces more complex form of something that always opposes the motion of an object is drag. Drag is when an object mores through a fluid (gas or liquid). In general this can be very complex in certain situations like a supersonic aircraft at mach 5 (5 time the speed of sound). However, we can deduce some basic ideas about drag. It is proportional to the area () of the object in the direction of motion. It is proportional to the density (mass per unit volume) of the fluid (ρ) and is proportional to the speed squared (v 2 ) of the object through the fluid. We can write this as an equation as: D = 1 2 Cρv2 where C is the empirically determined drag coefficient. Drag is what determines the terminal speed of a skydiver. If the skydiver is spreadeagle flat on to the wind this is normally about 200 km/hr. The much larger area of the open parachute reduces this to s safe speed for the person to land. let s do problem 520 Stoke s Law If an object is going slow, is very small or is in a medium much denser than air such as a liquid, the quadratic dependence on velocity does not hold. The object reaches terminal velocity very quickly. If the object is a sphere and the flow is laminar (nonturbulent), the drag force is given by Stoke s Law: S = 6πηrv where r is the radius of the sphere, η is the viscosity of the fluid and v is the velocity. Deformations Conceptually, the forces between atoms in an extended object can be considered to be tiny springs. When we deform an object by squeezing or stretching i.e., applying a force, the object will slightly change shape. When we release the force, the object returns to it s original shape. The more force you apply, the more the object will deform.
3 Hooke s Law Hooke s law describes deformation in one dimension when the object will return to its original state when the force is released. It is often associated with springs but many objects will behave in this way. Hooke s law is stated as = k L where is the force applied, L is the amount of deformation and k is the constant of proportionality knows as Hooke s constant. Hooke s constant depends on the material, the direction of the force and the material s geometry. K is normally determined empirically. Compression and Tension  Young s Modulus Conceptually, the forces between atoms in an extended object can be considered to be tiny springs. When we deform an object by squeezing or stretching i.e., applying a force, the object will slightly change shape. When we release the force, the object returns to it s original shape. The more force you apply, the more the object will deform. L o Lo + L If the amount of change is small, the relation between the dimensional changes and the force is given by: = Y L L where is the force, L is the change in length of the object, L the the original length of the object, is the object s cross sectional area and Y is the Young s modulus. Y depends on the material. table for some common material is given on page 177. The units of the Young s modulus is N/m 2. Let s work problems 529
4 Shear Deformation  Shear Modulus nother type of deformation is shear where the force twist the object. x L o The force necessary to make a deformation of x is given by: = S x L where S is called the shear modulus (page 177), x is the deformation, is the cross sectional area and L is the thickness of the object. The units of the shear modulus is N/m 2. Let s work problems 534 Bulk Deformation  Bulk Modulus It is also possible to change the volume of something by pressing on all sides with a uniform pressure. If we compress an object the relationship between pressure change, P, volume change V, and the original volume, V, is: P = B V V where B is the bulk modulus (see table on page 177). The units are N/m 2. Notice the area is now in the pressure so dimensionally this equation is the same as the shear and Young s modulus equations. Let s work problems 541 ll of this is rather empirical. It is not related to some fundamental principle, it is just an equation which describes the effect. We can deform an object past the point where it will return it its original shape. In this situation, the elastic limit has been exceeded for the material. ductile material(like many metals) will not suddenly break when too much force is applied but stretch out into a thinner cross section. The point where the stress on a material (/) will cause the material to deform is the elastic
5 limit. brittle material (like glass) will suddenly break. Some material will behave in different ways when a compression force is applied and another when a stretching force is applied. Concrete and stone can support huge forces when compressed but can be broken if too much tensile strength.
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