Example 1: You tie a rope to a tree and you pull on the rope with a force of 100 N. What is the tension in the rope?

The tension in the rope is the force that the rope “feels” across any section of it (or that you would feel if you replaced a piece of the rope). Since you are pulling with a force of 100 N, that is the tension in the rope.

Applications of Newton’s Laws (Examples)Ropes and tension

Example 2: Two tug-of-war opponents each pull with a force of 100 N on opposite ends of a rope. What is the tension in the rope?

This is literally the identical situation to the previous question. The tension is not 200 N !! Whether the other end of the rope is pulled by a person, or pulled by a tree, the tension in the rope is still 100 N !!

Example 3: You and a friend can each pull with a force of 20 N. If you want to rip a rope in half, what is the best way?1) You and your friend each pull on opposite ends of the rope2) Tie the rope to a tree, and you both pull from the same end3) It doesn’t matter -- both of the above are equivalent

Take advantage of the fact that the tree can pull with almost any force (until it falls down, that is!). You and your friend should team up on one end, and let the tree make the effort on the other end.

Example:

m M FfT

Given: m, M, f, FT-? a-?

F-T = Ma (1)

T-f = ma (2)

F-f=(M+m)a

T=F-Ma

mM

fFa

)( fFmM

MFT

A) M=m

;2m

fFa

2/)()(

2

1fFfFFT

B) M>>m

;M

fFa

fT

Example: Four blocks of mass m1=1kg, m2=2kg, m3=3kg, m4=4kg are on a frictionless horizontal surface as shown on the figure below. The blocks are connected by ideal massless strings. A force FL=30N is applied to the left block and is directed to the left. Another force FR=50N is applied to the right block, and is directed to

the right. What is the magnitude of the tension T in the string between m2 and m3. m1=1kg m2=2kg m3=3kg m4=4kg FR=50NFL=30N

T=?

NNNkg

kgNT

FFmmmm

mmFammFT

ammTF

NNNkg

kgNT

FFmmmm

mmFammFT

ammFT

smkg

NN

mmmm

FFa

ammmmFF

lRRR

R

lRll

l

lR

lR

36)3050(10

750

)(

)()(

)(

36)3050(10

330

)(

)()(

)(

/210

3050

)(

)(

4321

4343

43

4321

2121

21

2

4321

4321

1)

2a)

2b)

Newton’s eq. for all masses

Newton’s eq. for m1 and m2

Newton’s eq. for m3 and m4

In both of the cases depicted to the right, a 2-kg weight is supported by two strings. In which case are the students holding the strings exerting a force of greater magnitude?

1. Case A

2. Case B

3. Both the same

4. Need more information

Example: Sharing the weight.

A

B

W

W

F1 +F2

F1 +F2

F1F2

F1 F2

A

B

θ θ

mg

T T

Example: Find the tension in the cables.

: cos cos 0x T T

x

y

: 2 sin 0y T mg 2sinmg

T

Small θ, large T

m

It is impossible for a real cable (m > 0) to be completely horizontal (it would require infinite tension, and then the cable snaps).

M=0

m1 m2

m1gm2g

TT

ammgmm

amgmT

amTgm

2121

22

11

Atwood’s Machine

gmm

mma

21

21

gmTgm

mm

gmmagmT

21

21

212

2

Pulleys

Example: In which case does block m experience a larger acceleration? In (1) there is a 10 kg mass hanging from a rope and falling. In (2) a hand is providing a constant downward force of 98 N. Assume massless ropes.

m

M=10kg a

m

a

F = 98 N

Case (1) Case (2)

In (2) the tension is T2 = 98 N due to the hand.

In (1) the tension T1 is less than 98 N because the block is accelerating down.

Only if the block were at rest would the tension be equal to 98 N.

Because of that in case 2 block m experience a larger acceleration.

211 )( TMgagMTMaTMg

MgNT 982

Example: How much force does the worker have in order to support the mass M at constant height h off the ground?

A. mg; B. mg/2; C. 2mg; D. 3mg; E. mg/3

TT

T1

TT

T2

T1

mg

mgTT

mgTT

mgT

2

2/2/

2

1

1

T

mg=6T