Friction

Nf kk

Nf ss

Kinetic friction:

fk- friction forceN - normal force

k - coefficient of kinetic friction

Static friction:

- coefficient of static frictionssk

Fbook,hand Nbook,wall

fS,book,wall

Wbook,Earth

For the book not to fall down, fS = W

Pushing harder (increasing Fbook,hand) increases Nbook,wall and therefore fS,MAX increases, but not the actual value of fS that we had, which needs to continue to be exactly W.

Example: When you push a book against a wall, the static friction between the wall and the book can prevent it from falling.

If you press harder, the friction force will be:

A. Larger than before

B. The same

C. Smaller than before.

Example: A 49 kg rock climber is climbing a “chimney” between two rock slabs. The coefficient of static friction between her shoes and the rock is 1.2; between her back and the rock it is 0.80. She has reduced her push against the rock until her back and her shoes are on the verge of slipping. What is her push against the rock?

W

N

Nfs backfs feet

x

y

N

fS backfS feet

N

w

0 :

0 :

,,

yfeetSbackS

x

mamgffy

maNNx

If the climber dos not slip or move up, the value of the sum of frictional forces is fixed:

mgff feetSbackS ,,

S SS,back S,f eet

S

with

I f is large enough, both f orces will be than

ff mg f N

N N

less

S,back S,f eet

2

S,back S,f eet

0

(49 kg)(9.8 m/ s )240 N

( ) 1.2+0.80

N N mg

mgN

S S

As decreases, the maximum static f riction gets closer

to the actual value of f riction.

Eventually, is just enough so that . I f the

climber pushes less than this, the f riction will not be

enou

N

N f N

gh to compensate her weight.

Answer B

Fby you fS by floorfk by floor

Fby you

friction

μsN

μkN

Static friction Kinetic friction

Example: Trying to move a trunk

Fby you

fS by floor

You increase the applied force, but the trunk still does not move: static friction is increasing too.

Example: Trying to move a trunk

Fby you

For “small” forces, the trunk does not move. So there must be a friction force fS = Fby you fS by floor

Fby you

fS by floorfk by floor

You increase the applied force even more. Eventually, the trunk moves. Static friction cannot be larger than a certain value.

And then friction becomes kinetic!

Example: Box on incline with friction

yW xW

N

W

y

x

f

cos

sin

mgW

mgW

mgW

y

x

mafmg

mgN

sin

0cos

criticals

s

s

s

sss

mgmg

mgf

mgNf

a

tan

tan

cossin

sin

cos

0

1)Static:

2) Kinetic

)cos(sin

cossin

cos

k

k

kkk

ga

mamgmg

mgNf

mgsinθ – F = m ax = 0

N – mgcosθ = m ay = 0

F = mg sinθ

N = mg cosθ

mg

x

y

θθ

N

FB,hand

Example: A hand keeps a 20-kg box from sliding down a frictionless incline. The plane of the incline makes an angle θ = 30° with the horizontal. What is the magnitude of the force exerted by hand?

mgsinθ

mgcosθ

F = (20 kg)(9.8 m/s2)sin(30°) = 98 N

Directions:

•Draw the free-body diagram

• Choose axes (draw them!)

• Use Newton’s 2nd law in the x and y-directions.

Centripetal force

F = ma rT

mrmr

mvF

222

Example: A stone of mass m sits at the bottom of a bucket. A string is attached to the bucket and the whole thing is made to move in circles. What is the minimum speed that the bucket needs to have at the highest point of the trajectory in order to keep the stone inside the bucket?

F = mar

mvF

2

r

mvmgN

2

mgr

mvN

2

grv min0min N

Examples (centripetal force)

(3)

(2)

(1) mgN

r

mvmgN

2

r

mvmgN

2

mgr

mvN

2

mgr

mvN

2

r

mvmgN

2

r

mvNmg

2

mg

N

N

mg

F = mar

mvF

2

Example: Little Jacob (15 kg) sits on the edge of a merry-go-round of radius 1.0 m while big sister makes it turn… faster and faster. How fast can the system go before Jacob takes off if the coefficient of static friction between Jacob’s pants and the merry-go-round is 0.5?

mg

N

fs

Static friction provides the needed radial acceleration:2

Sf mr

Maximum speed Maximum static friction:2MAXS mg mr

MAXS gr

2(0.5)(9.8 m/ s ) 12.2 rad/ s turn/ s

1 m 3

Example: A car of mass m with constant speed v drives through a curve of radius R. What is the minimum value of the coefficient of static friction between the tires and the road for the car not to slip?

s S Sf N mg netF ma

2

S

vm mg

R

2

s

vf m

R

2

S

vgR

fs