3Forces

IA Dynamics and Relativity

3.6 Friction

At an atomic level, energy is always conserved. However, in many everyday

processes, this does not appear to be the case. This is because friction tends to

take kinetic energy away from objects.

In general, we can divide friction into dry friction and fluid friction.

Dry friction

When solid objects are in contact, a normal reaction force

N

(perpendicular

to the contact surface) prevents them from interpenetrating, while a frictional

force

F

(tangential to the surface) resists relative tangential motion (sliding or

slipping).

N

F

mg

If the tangential force is small, it is insufficient to overcome friction and no

sliding occurs. We have static friction of

|F| ≤ µ

s

|N|,

where µ

s

is the coefficient of static friction.

When the external force on the object exceeds

µ

s

|N|

, sliding starts, and we

have a kinetic friction of

|F| = µ

k

|N|,

where µ

k

is the coefficient of kinetic friction.

These coefficients are measures of roughness and depend on the two surfaces

involved. For example, Teflon on Teflon has coefficient of around 0.04, while

rubber on asphalt has about 0.8, while a hypothetical perfectly smooth surface

has coefficient 0. Usually, µ

s

> µ

k

> 0.

Fluid drag

When a solid object moves through a fluid (i.e. liquid or gas), it experiences a

drag force.

There are two important regimes.

(i)

Linear drag: for small things in viscous fluids moving slowly, e.g. a single

cell organism in water, the friction is proportional to the velocity, i.e.

F = −k

1

v.

where

v

is the velocity of the object relative to the fluid, and

k

1

>

0 is a

constant. This

k

1

depends on the shape of the object. For example, for a

sphere of radius R, Stoke’s Law gives

k

1

= 6πµR,

where µ is the viscosity of the fluid.

(ii)

Quadratic drag: for large objects moving rapidly in less viscous fluid, e.g.

cars or tennis balls in air, the friction is proportional to the square of the

velocity, i.e.

F = −k

2

|v|

2

ˆ

v.

In either case, the object loses energy. The power exerted by the drag force is

F · v =

(

−k

1

|v|

2

−k

2

|v|

3

Example.

Consider a projectile moving in a uniform gravitational field and

experiencing a linear drag force.

At t = 0, we throw the projectile with velocity u from x = 0.

The equation of motion is

m

dv

dt

= mg − kv.

We first solve for v, and then deduce x.

We use an integrating factor exp(

k

m

t) to obtain

d

dt

e

kt/m

v

= e

kt/m

g

e

kt/m

v =

m

k

e

kt/m

g + c

v =

m

k

g + ce

−kt/m

Since v = u at t = 0, we get c = u −

m

k

g. So

v =

˙

x =

m

k

g +

u −

m

k

g

e

−kt/m

.

Integrating once gives

x =

m

k

gt −

m

k

u −

m

k

g

e

−kt/m

+ d.

Since x = 0 at t = 0. So

d =

m

k

u −

m

k

g

.

So

x =

m

k

gt +

m

k

u −

m

k

g

(1 − e

−kt/m

).

In component form, let x = (x, y), u = (u cos θ, u sin θ), g = (0, −g). So

x =

mu

k

cos θ(1 − e

−kt/m

)

y = −

mgt

k

+

m

k

u sin θ +

mg

k

(1 − e

−kt/m

).

We can characterize the strength of the drag force by the dimensionless constant

ku/(mg), with a larger constant corresponding to a larger drag force.

Effect of damping on small oscillations

We’ve previously seen that particles near a potential minimum oscillate indefi-

nitely. However, if there is friction in the system, the oscillation will damp out

and energy is continually lost. Eventually, the system comes to rest at the stable

equilibrium.

Example.

If a linear drag force is added to a harmonic oscillator, then the

equation of motion becomes

m

¨

x = −mω

2

x − k

˙

x,

where

ω

is the angular frequency of the oscillator in the absence of damping.

Rewrite as

¨

x + 2γ

˙

x + ω

2

x = 0,

where γ = k/2m > 0. Solutions are x = e

λt

, where

λ

2

+ 2γλ + ω

2

= 0,

or

λ = −γ ±

p

γ

2

− ω

2

.

If

γ > ω

, then the roots are real and negative. So we have exponential decay.

We call this an overdamped oscillator.

If 0

< γ < ω

, then the roots are complex with

Re

(

λ

) =

−γ

. So we have

decaying oscillations. We call this an underdamped oscillator.

For details, refer to Differential Equations.