University Physics: Mechanics

Ch6. Friction

Lecture 9

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

9/2Erwin Sitompul University Physics: Mechanics

The figure below shows a coin of mass m at rest on a book that has been tilted at an angle θ with the horizontal. By experimenting, you find that when θ is increased to 13°, the coin is on the verge of sliding down the book, which means that even a slight increase beyond 13° produces sliding.

What is the coefficient of static friction μs between the coin and the book?Hint: Draw the free-body diagram of the coin first.

Homework 7: Coin On A Book

9/3Erwin Sitompul University Physics: Mechanics

Forces along the y axis:

Forces along the x axis:

net,y yF ma

net,x xF ma

N g cos 0F F

ssin 0gF f

N g cosF F

s Nsin 0gF F

N cosF mg

ssin cos 0mg mg

s

sin

cos

tan

• Why zero?

• Why zero?

So, the coefficient of static friction is:

s tan13 0.231

h

d

Solution of Homework 7: Coin On A Book

9/4Erwin Sitompul University Physics: Mechanics

A block of mass m = 3 kg slides along a floor while a force F of magnitude 12 N is applied to it at an upward angle θ. The coefficient of kinetic friction between the block and the floor is μk = 0.4. We can vary θ from 0 to 90° (with the block remains on the floor. What θ gives the maximum value of the block’s acceleration magnitude a?

→

Example: Blue Block

9/5Erwin Sitompul University Physics: Mechanics

Forces along the y axis: Forces along the x axis:

net,y yF ma

net,x xF ma

N g 0yF F F

N sinF mg F kxF f ma

k NcosF F ma

kcos sinF F

a gm m

• What θ gives the maximum value of a?

• da/dθ = 0

Example: Blue Block

9/6Erwin Sitompul University Physics: Mechanics

kcos sinF F

a gm m

ksin cosda F F

d m m

0

ktan 1

ktan

then, the derivative of a with respect to θ is

If a is given by

1tan (0.4)

21.80

Example: Blue Block

9/7Erwin Sitompul University Physics: Mechanics

Block B in the figure below weighs 711 N. The coefficient of static friction between block and table is 0.25; angle θ is 30°. Assume that the cord between B and the knot is horizontal. Find the maximum weight of block A for which the system will be stationary.

Example: Two Blocks

9/8Erwin Sitompul University Physics: Mechanics

fs,max

→

Block B

Block A

Knot

TB

→

TA

→

TB

→

TA

→

Wall

TW

→

TW

→

FgA

→

Knot

TW

→

fs,max

→

FgA

→

FNB

→

FgB

→

Example: Two Blocks

9/9Erwin Sitompul University Physics: Mechanics

Knot

TW

→

fs,max

→

FgA

→

TWx

TWyForces along the y axis:

Forces along the x axis:

net, 0yF

net, 0xF

W gA 0yT F

W s,max 0xT f

W AsinT m g θ

W s NBcosT F

W s BcosT m g

s BA

sin cos

m gm g

s BA

sin cos

WW

102.624 N

(0.25)(711) tan 30

A s B tanW W

Example: Two Blocks

9/10Erwin Sitompul University Physics: Mechanics

A block of mass m1 on a rough, horizontal surface is connected to a ball of mass m2 by a lightweight cord over a lightweight, frictionless pulley as shown in the figure below.A force of magnitude F at an angle θ with the horizontal is applied to the block as shown and the block slides to the right. The coefficient of kinetic friction between the block and surface is μk. Find the magnitude of acceleration of the two objects.

Example: Multiple Objects

9/11Erwin Sitompul University Physics: Mechanics

N g1 yF F F 2 ( )T m g a

k N 1cosF T F m a

fk

→ m1

m2

T→

T→

Fg2

→

FN

→

Fg1

→

F→

Fx

Fy

θ

Forces in m2

net, 2 2y yF m a

g2 2T F m a

Forces in m1

net, 1 1x xF m a

k 1xF T f m a

1 k NcosT F m a F

net, 0yF

N g1 0yF F F

N 1 sinF m g F

Example: Multiple Objects

9/12Erwin Sitompul University Physics: Mechanics

N 1 sinF m g F

2 ( )T m g a

1 k NcosT F m a F

2 1 k 1( ) cos ( sin )m g a F m a m g F

1 2 k k 1 2cos sinm a m a F F m g m g

1 2 k k 1 2( ) (cos sin ) ( )m m a F m m g

k k 1 2

1 2

(cos sin ) ( )F m m ga

m m

Example: Multiple Objects

9/13Erwin Sitompul University Physics: Mechanics

When the three blocks in the figure below are released from rest, they accelerate with a magnitude of 0.5 m/s2. Block 1 has mass M, block 2 has 2M, and block 3 has 2M.What is the coefficient of kinetic friction between block 2 and the table?

Example: Trio Blocks

9/14Erwin Sitompul University Physics: Mechanics

1 ( )T M g a

2 2 ( )T M g a

FN

→

fk

→

T1

→

m1

T1

→

Fg1

→

m2

Fg2

→

T2

→

aa

a

m3

T2

→

Fg3

→

Forces in m1

net, 1 1y yF m a

1 g1T F Ma

Forces in m3

net, 3 3y yF m a

2 g3 2 ( )T F M a

N 2F Mg

Forces in m2

net, 2 2y yF m a

N g2 0F F

2 1 k N 2T T F Ma

net, 2 2x xF m a

2 1 k 2T T f Ma

Example: Trio Blocks

9/15Erwin Sitompul University Physics: Mechanics

1 ( )T M g a

2 2 ( )T M g a N 2F Mg2 1 k N 2T T F Ma k2 ( ) ( ) 2 2M g a M g a Mg Ma

k 2 2 ( ) ( ) 2Mg M g a M g a Ma

k

2 ( ) ( ) 2

2

M g a M g a Ma

Mg

k

5

2

Mg Ma

Mg

5

2

g a

g

(9.8) 5(0.5)

2(9.8)

20.372 m s

Example: Trio Blocks

9/16Erwin Sitompul University Physics: Mechanics

End of the Class