Friction & Air Resistance

whenever an object moves or attempts to move on a surface, there is a force

parallel to the surface in the opposite direction to the motion or attempt at motion

• this is known as friction

if an object is attempting to move, the frictional force is

static friction

if an object is already moving, the frictional force is

kinetic friction

static friction

if an object is pushed with an increasing force, the force of static friction fs will increase to exactly balance the applied force

• the object thus does not accelerate, and the acceleration = 0 and thus Fnet = 0 by Newton’s 2nd law of Fnet = ma

• the static friction force fs is equal and opposite to the applied force Fa if the object does not move

at fs,max, the acceleration of the object is

a > 0 and thus Fnet > 0, and the object is still at rest and thus its velocity v = 0, but it is about to move

• an object will begin to slide if the applied force is increased beyond fs,max

when the applied force exceeds the maximum possible static friction fs,max, the object will

begin to move from rest (v = 0)

• a > 0, Fnet > 0

static friction force and coefficient

• for an object that is not slipping: fs = -Fa, and fs < fs,max

• for an object that is on the verge of slipping: fs = fs,max = us N

  • fs,max is proportional to the normal force (N) that the bottom surface exerts on the top surface

  • us is the proportionality constant of static friction

origin of friction occurs from

a few contact points between the top and bottom surface which hinders the sliding of two objects against each other

• a larger force needs to be applied to overcome the contact points, and the magnitude of the force depends on the height of the peaks and valleys (roughness of the surfaces)

the coefficient of static friction, us, depends on the

roughness of the two surfaces in contact

• it is thus a ratio between two surfaces, and is dimensionless

• ex. wood on wood, rubber on asphalt, metal on ice

when fs < fs,max, the static friction does not depend on

the static friction coefficient us at all

• the coefficient us only matters when the object is on the verge of slipping, and thus fs = fs,max

kinetic friction fk is the

frictional force parallel to the surface that opposes the motion of a moving object

• fk = uk N, and thus the force of kinetic friction fk is proportional to the normal force (N)

• the kinetic friction coefficient uk depends on the roughness of the two surfaces in temporary contact

• independent of the pushing / pulling force Fa

  • fs,max is greater than fk, which is constant (fk < fs,max)

  • uk < us

it is generally harder to make an object slide starting from

rest than it is to keep making it slide

the force of kinetic friction, fk, is

• independent of the contact area between surfaces

• approximately independent of the speed, as long as the speed is not too large

air resistance (drag)

always points in the opposite direction of motion and is dependent on velocity / speed

• its magnitude is Fd = ½ CApv²

  • C: streamline coefficient (shape dependent)

  • A: cross-sectional area of the object

  • p: density of air (1 kg / m³)

  • v: speed of the object

motion in the presence of air resistance is

not at a constant acceleration, but it is at the very acceleration until it reaches the terminal velocity, from which point on it falls with constant velocity (a = 0)

• this is when the force of air resistance (Fd) matches the gravitational force of the Earth on the object (mg)

  • Fnet = 0 (equal and opposite forces) and thus a = 0 (constant terminal velocity)

terminal velocity

any object falling in air will eventually reach a terminal velocity when air resistance is equal and opposite to the gravitational force driving the fall

• applying Newton’s 2nd law: ½ CApvt² - mg = 0, thus vt² = 2mg / CAp, and thus vt = √2mg / CAp

terminal velocity vt is proportional to

√m and inversely proportional to √A

• if the cross-sectional area of the object is increased, the terminal velocity is decreased

• if the mass of the object is increased, the terminal velocity will be reached later during the fall and will therefore have a larger magnitude