Section Summary

Section Summary

5.1 Friction

  • Friction is a contact force between systems that opposes the motion or attempted motion between them. Simple friction is proportional to the normal force NN size 12{N} {} pushing the systems together. A normal force is always perpendicular to the contact surface between systems. Friction depends on both of the materials involved. The magnitude of static frictionfsfs size 12{f rSub { size 8{s} } } {} between systems stationary relative to one another is given by
    fsμsN,fsμsN, size 12{f rSub { size 8{s} } <= μ rSub { size 8{s} } N} {}
    where μsμs size 12{μ rSub { size 8{s} } } {} is the coefficient of static friction, which depends on both of the materials.
  • The kinetic friction force fkfk size 12{f rSub { size 8{k} } } {} between systems moving relative to one another is given by
    fk=μkN,fk=μkN, size 12{f rSub { size 8{k} } =μ rSub { size 8{k} } N} {}
    where μkμk size 12{μ rSub { size 8{K} } } {} is the coefficient of kinetic friction, which also depends on both materials.

5.2 Drag Forces

  • Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity vv in air, the drag force is given by
    FD=12CρAv2,FD=12CρAv2, size 12{F rSub { size 8{D} } = { {1} over {2} } Cρ ital "Av" rSup { size 8{2} } } {}
    where CC size 12{C} {} is the drag coefficient (typical values are given in Table 5.2), AA size 12{A} {} is the area of the object facing the fluid, and ρρ size 12{ρ} {} is the fluid density.
  • For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes' law,
    Fs=6πηrv,Fs=6πηrv, size 12{F rSub { size 8{D} } =6 ital "πη" ital "rv"} {}
    where rr size 12{r} {} is the radius of the object, ηη size 12{η} {} is the fluid viscosity, and vv size 12{v} {} is the object's velocity.

5.3 Elasticity: Stress and Strain

  • Hooke's law is given by
    F=kΔL,F=kΔL, size 12{F=kΔL} {}

    where ΔLΔL size 12{ΔL} {} is the amount of deformation (the change in length), FF size 12{F} {} is the applied force, and kk size 12{k} {} is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as

    ΔL=1YFAL0,ΔL=1YFAL0, size 12{ΔL= { {1} over {Y} } { {F} over {A} } L rSub { size 8{0} } } {}

    where YY size 12{Y} {} is Young's modulus (which depends on the substance), AA size 12{A} {} is the cross-sectional area, and L0L0 size 12{L rSub { size 8{0} } } {} is the original length.

  • The ratio of force to area, FAFA size 12{ { {F} over {A} } } {}, is defined as stress, measured in N/m2.
  • The ratio of the change in length to length, ΔLL0ΔLL0 size 12{ { {ΔL} over {L rSub { size 8{0} } } } } {}, is defined as strain (a unitless quantity). In other words,
    stress=Y×strain.stress=Y×strain. size 12{"stress"=Y times "strain"} {}
  • The expression for shear deformation is
    Δx=1SFAL0,Δx=1SFAL0, size 12{Δx= { {1} over {S} } { {F} over {A} } L rSub { size 8{0} } } {}

    where SS is the shear modulus and FF is the force applied perpendicular to L0L0 and parallel to the cross-sectional area AA.

  • The relationship of the change in volume to other physical quantities is given by
    ΔV=1BFAV0,ΔV=1BFAV0, size 12{ΔV= { {1} over {B} } { {F} over {A} } V rSub { size 8{0} } } {}

    where BB is the bulk modulus, V0V0 is the original volume, and FAFA size 12{ { {F} over {A} } } {} is the force per unit area applied uniformly inward on all surfaces.