LECTURE 12INTERCONNECTING OBJECTS, ROPES AND PULLEYS

Traction saved many lives during World War II by allowing soldiers to be transported safely without injury to their surrounding tissues.

5.7 Interconnecting objects Objects in contact

5.6 Ropes and pulleysPulleys

Learning objectives

! Apply Newton’s laws to a system of interacting objects as a whole and to each individual object within the system.

! Identify Newton’s 3rd law pairs within a system of interacting objects.! Use the massless approximation for ropes to calculate tension.! Calculate tensions in a rope and pulley system.

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Quiz: 5.7-1

! Two blocks with different masses (labeled 1 and 2) are in contact on a frictionless table. A horizontal force of !"# is applied to block 1 by a hand. Which of the following statements is/are correct? Choose all that apply.A. The acceleration of block 1 is the same as that of block 2.B. The net force on block 1 is the same as the net force on block 2.C. For each block, the magnitude of the normal force by the table is equal to the magnitude

of the weight of the block.D. The magnitude of the normal force by block 1 on block 2 is equal to the magnitude of the

normal force by block 2 on block 1.

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21!"#

Quiz: 5.7-1 answer

A. The acceleration of block 1 is the same as that of block 2.! Since the blocks are in contact and move together, they must share the same acceleration.

B. The net force on block 1 is the same as the net force on block 2.! Since they have different masses, if they have the same acceleration, they must have different net forces on them.

"⃗#$% = '(⃗ (Newton’s 2nd law)

C. For each block, the magnitude of the normal force by the table is equal to the magnitude of the weight of the block.

! Since the blocks are not accelerating vertically, the net force in the vertical direction is zero for either block. Only forces in the vertical direction are weight and the normal force on the block by the floor.

D. The magnitude of the normal force by block 1 on block 2 is equal to the magnitude of the normal force by block 2 on block 1.

! Newton’s 3rd law pair.

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21")*

Quiz: 5.7-2

! Two blocks with different masses (labeled 1 and 2) are in contact on a frictionless table. A horizontal force of !"# is applied to block 1 by a hand. Rank the following magnitudes of forces, smallest first.A. !"#B. !#"C. $#%D. $%#

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21!"#

Quiz: 5.7-2 answer

! Two blocks with different masses (labeled 1 and 2) are in contact on a frictionless table. A horizontal force of !"# is applied to block 1 by a hand. Rank the following magnitudes of forces, smallest first.

! $#% = $%# < !"# = !#"! Newton’s 3rd law pair: $#% = $%#! Newton’s 3rd law pair: !"# = !#"! Block 1 accelerates to the right, so net force must be to the right: !"# > $%#

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21!"#

FBD of block 1(horizontal forces only)

!⃗"#$%#

Quiz: 5.7-3

! Two blocks (!" = 2.3 kg and !$ = 1.2 kg) are in contact on a frictionless table. A horizontal force of %"& = 3.2 N is applied to block 1 by a hand. Find the magnitude of the acceleration of the two blocks, ', in m/s2.

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!$!"%&"

Quiz: 5.7-3 answer

! Two blocks (!" = 2.3 kg and !$ = 1.2 kg) are in contact on a frictionless table. A horizontal force of %"& = 3.2 N is applied to block 1 by a hand. Find the magnitude of the acceleration of the two blocks, ', in m/s2.

! The net force on the two-block system is 3.2 N. So the acceleration of the system is

' = ()*+, -.-+*/0-.-+*/

= 1.$ 31.4 56 = 0.91 m/s$.

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!$!"

%&"

FBD of system(horizontal force only)

%⃗&"

Quiz: 5.7-4

! Two blocks (!" = 2.3 kg and !$ = 1.2 kg) are in contact on a frictionless table. A horizontal force of %"& = 3.2 N is applied to block 1 by a hand. Find the magnitude of the normal force in N between the two blocks, '.

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!$!"%&"

Quiz: 5.7-4 answer

! Two blocks (!" = 2.3 kg and !$ = 1.2 kg) are in contact on a frictionless table. A horizontal force of %"& = 3.2 N is applied to block 1 by a hand. Find the magnitude of the normal force in N between the two blocks, '.

! The acceleration of the two-block system is ( = 0.91 m/s$.! Block 1 alone also accelerates at 0.91 m/s$ while the net force on it is 3.2 N − '$" = !"(.

! From Newton’s 3rd law, ' = '$" = '"$ = 3.2 N −!"( = 3.2 N − 2.3 kg 0.91 678 =1.1 N

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!$!"

%&"

FBD of block 1(horizontal forces only)

%⃗&"'$"

5.8 Ropes

! Generally, the tension in a massless string or rope equals the magnitude of the force pulling on the end of the string or rope.

! The tension in a massless string or rope is the same from one end to the other.

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Quiz: 5.8-1

! All three 50-kg blocks are at rest. Assume that the ropes are massless. Rank the magnitude of the tensions, smallest first.

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Quiz: 5.8-1 answer

!

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! All three 50-kg blocks are at rest. Assume that the ropes are massless. Rank the magnitude of the tensions, smallest first.

! "# = "% = "& = 50 kg 9.8 ⁄m s%! The force exerted by the frictionless massless pulley on the rope

is always a normal force, which has no component tangent to the rope, so it cannot produce a change in the magnitude of tension.

Quiz: 5.8-2

! A boy sits in a tire that is attached to a massless rope that passes over a massless frictionless pulley fastened to the ceiling and then passes back down to the boy's hands. The weight of the boy plus the tire is ! ≡ !#$. What is the force with which the boy must pull down on the free end of the rope to support his weight in the tire in terms of !?

Quiz: 5.8-2 answer

!!"

! According to Newton’s 3rd law, the boy is pulling on the rope as hard as the tension in the rope that is pulling his hand.

! Since the rope is massless, the magnitude of the tension throughout the rope is constant.

! All the forces applied on the boy/tire system are! Tensions in the rope upward � 2! Weight of the system, #

! The boy/tire system is not accelerating, so the net force on it is zero.

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FBD of boy/tire

$%& $%&

'(&

Quiz: 5.8-3

! In the block-and-pulley arrangement shown, three segments of the single rope pull on the block. Assume that the masses of the pulleys, the rope, and the chain are negligible. Assume that the segments of the rope are all vertical.

! What is the magnitude !"# of the force exerted by a person on the rope to raise the block at constant speed in terms of the mass of the block, $?

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!"#

Quiz: 5.8-3 answer / demo

! What is the magnitude !"# of the force exerted by a person on the rope to raise the block at constant speed in terms of the mass of the block, $?

! Since the block is moving at constant speed, so are the lower pulley and the chain.

! If you consider the system of the lower pulley, the block and the chain connecting them, the net force on it must be zero: 3&#' = )*', or &#' = +

,)*'! The magnitude of the tension is the rope is constant throughout the rope.

! So the tension that is pulling up on the person’s hand is also &#- = &#" = +,).*

! From Newton’s 3rd law, ! = &#- = +,).* =

+,$/

! Note that to pull up the block by some distance, the hand has to pull a longer distance on the rope.

! Demo: Pulleys

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!"#

FBD of lower pulley/chain/block

3&#"

).*

Applications of pulley system

! Rock climbing / rescue! Exercise machine! Traction is used primarily as a temporary measure

until corrective surgery is performed.

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