U4: Further Application of Newton's Laws

Friction is a force that opposes relative motion between systems in contact

● It is parallel to the contact surface between systems and always in a direction that opposes motion or attempted motion of the systems relative to each other

● If two systems are in contact and moving relative to one another, the friction between them is called kinetic friction

○ Ex: A hockey puck slowing down on ice

● When objects are stationary, static friction can act between them

○ Ex: Pushing a heavy crate that won’t move

○ The static friction is usually greater than the kinetic friction between objects

The magnitude of static friction is 𝑓s ≤ 𝜇s𝑁

● 𝜇s is the coefficient of static friction

● N is the magnitude of the normal force

● Once the applied force exceeds 𝑓s(max), the object will move

○ 𝑓s(max) = 𝜇s𝑁

The magnitude of kinetic friction is 𝑓k = 𝜇k𝑁

● 𝜇k is the coefficient of kinetic friction

● N is the magnitude of the applied force

The coefficients of static and kinetic friction depend on the system the objects are acting upon:

The drag force is found to be proportional to the square of the speed of the object

● It depends on the shape of the object, its size, its velocity, and the fluid it is in

● 𝐹𝐷 = .5𝐶𝜌𝐴𝑣2 

○ C is the drag coefficient

○ A is the area of the object facing the fluid

○ v is the speed of the object

○ 𝜌 is the density of the fluid

● Using Newton’s second law, we can determine that at the terminal velocity 𝑚𝑔 = 𝐹𝐷

○ So, 𝑚𝑔 = .5𝐶𝜌𝐴𝑣2 

Stokes’ Law states that 𝐹𝑠 = 6𝜋𝑟𝜂𝑣

● r is the radius of the object

● 𝜂 is the viscosity of the fluid

● v is the object’s velocity

Deformation is a change in shape due to the application of a force

● Even very small forces are known to cause some deformation

● Hooke’s Law states that 𝐹 = 𝑘𝛥𝐿

○ 𝛥𝐿 is the amount of deformation (like change in length) produced by F

○ k is a proportionality constant that depends on the shape and composition of the object and the direction of the force

■ The proportionality constant depends on a number of factors for the material Stress is the ratio of force to area

● Stress = F/A

● Strain is the ratio of the change in length to length

○ Strain = 𝛥𝐿/𝐿0

● Shear deformation behaves similarly to tension and compression and can be described with similar equations

○ 𝛥𝑥 = (I𝐹)/(𝑆𝐴)𝐿0

■ S is the shear modulus

■ F is the force applied perpendicular to 𝐿0 and parallel to the cross-sectional area A