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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 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 between systems stationary relative to one another is given by
    where μsμs is the coefficient of static friction, which depends on both of the materials.
  • The kinetic friction force fkfk between systems moving relative to one another is given by
    f k = μ k N , f k = μ k N ,
    where μkμk 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 v v in air, the drag force is given by
    F D = 1 2 CρAv 2 , F D = 1 2 CρAv 2 ,
    where CC is the drag coefficient (typical values are given in Table 5.2), AA is the area of the object facing the fluid, and ρρ 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,
    where rr is the radius of the object, ηη is the fluid viscosity, and vv is the object’s velocity.

5.3 Elasticity: Stress and Strain

  • Hooke’s law is given by

    where ΔLΔL is the amount of deformation (the change in length), FF is the applied force, and kk 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


    where YY is Young’s modulus, which depends on the substance, AA is the cross-sectional area, and L0L0 is the original length.

  • The ratio of force to area, FAFA, is defined as stress, measured in N/m2.
  • The ratio of the change in length to length, ΔLL0ΔLL0, is defined as strain (a unitless quantity). In other words,
  • The expression for shear deformation is

    where S S is the shear modulus and F F is the force applied perpendicular to L 0 L 0 and parallel to the cross-sectional area A A.

  • The relationship of the change in volume to other physical quantities is given by

    where B B is the bulk modulus, V 0 V 0 is the original volume, and FAFA is the force per unit area applied uniformly inward on all surfaces.

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