Section Summary

 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 byfs≤μsN,𝑓s≤𝜇s𝑁,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 byfk=μkN,𝑓k=𝜇k𝑁,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 v𝑣 in air, the drag force is given byFD=12CρAv2,𝐹D=12CρAv2,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,Fs=6πηrv,𝐹s=6πηrv,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 byF=kΔL,𝐹=𝑘Δ𝐿,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 asΔL=1YFAL0,Δ𝐿=1𝑌𝐹𝐴𝐿0,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,stress=Y×strain.stress=𝑌×strain.
• The expression for shear deformation isΔx=1SFAL0,Δ𝑥=1𝑆𝐹𝐴𝐿0,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 byΔV=1BFAV0,Δ𝑉=1𝐵𝐹𝐴𝑉0,where B𝐵 is the bulk modulus, V0𝑉0 is the original volume, and FA𝐹𝐴 is the force per unit area applied uniformly inward on all surfaces.