College Physics for AP® Courses

# Section Summary

### 5.1Friction

• 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 friction$fsfs size 12{f rSub { size 8{s} } } {}$ between systems stationary relative to one another is given by
$fs≤μsN,fs≤μsN, size 12{f rSub { size 8{s} } <= μ rSub { size 8{s} } N} {}$
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
$f k = μ k N , f k = μ k N , size 12{f rSub { size 8{k} } =μ rSub { size 8{k} } N} {}$
where $μkμk size 12{μ rSub { size 8{K} } } {}$ is the coefficient of kinetic friction, which also depends on both materials.

### 5.2Drag 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
$F D = 1 2 CρAv 2 , F D = 1 2 CρAv 2 , size 12{F rSub { size 8{D} } = { {1} over {2} } Cρ ital "Av" rSup { size 8{2} } } {}$
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.

### 5.3Elasticity: 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 $S S$ 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
$ΔV=1BFAV0,ΔV=1BFAV0, size 12{ΔV= { {1} over {B} } { {F} over {A} } V rSub { size 8{0} } } {}$

where $B B$ is the bulk modulus, $V 0 V 0$ is the original volume, and $FAFA size 12{ { {F} over {A} } } {}$ is the force per unit area applied uniformly inward on all surfaces.