Work done by a spring

1
(No Transcript)
2

• Work done by a spring
• Force exerted by a spring always tries to restore
  a spring to its
• neutral position

U12 ½(x12 x22)k Note if x1 is not the neutral
position U12 is not equal to -(k/2)Dx2
U12 T2 T1 or T1 U12 T2
3

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block.

4

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft

5

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure

6

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity

7

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B

8

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02

9

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2)

10

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)

11

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2

12

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0

13

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• . .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft

.5
14

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 1.86 ft/s

15

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 3.48 ft/s
• TA UAC TC

16

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 3.48 ft/s
• TA UAC TC
• .5(.5/g)02

17

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 3.48 ft/s
• TA UAC TC
• .5(.5/g)02 (9/2)(.1332 02) 0.5(y .133)

18

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 3.48 ft/s
• TA UAC TC
• .5(.5/g)02 (9/2)(.1332 02) 0.5(y .133)
  .5(.5/g)02

19

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 3.48 ft/s
• TA UAC TC
• .5(.5/g)02 (9/2)(.1332 02) 0.5(y .133)
  .5(.5/g)02
• y .0266 ft or ytotal .1333 .0266 .1599 ft

20

• Work Energy Problems With Springs
• Problem 13.25 A 0.7 lb block rests on top of a
  0.5 lb
• block supported by but not attached to a spring
  of
• constant 9 lb/ft. The upper block is suddenly
  removed.
• Determine (a) the maximum velocity reached by the
  
• 0.5 lb block, (b) the maximum height reached by
  the
• 0.5 lb block. Solution find the initial
  compression in the
• Spring FkDy ?Dy 1.2/9 0.133 ft
• Put a datum line on the figure
• TA UAB TB where A is the initial position
  and
• B is the Position
  with maximum velocity
• energy at A work from A to B energy at B
• .5(.5/g)02 (9/2)(.1332 y2) 0.5(y .133)
  .5(.5/g)vB2
• vB (4g(.013 - .5y - (9/2)y2))1/2 for maximum
  dvB/dy 0
• dvB/dy (1/2)(4g (.08 - .5y - (9/2)y2))-1/24g(-.5
  9y) 0
• -.5 - 9y 0 ? y -.0555 ft ? vB 3.48 ft/s
• TA UAC TC
• .5(.5/g)02 (9/2)(.1332 02) 0.5(y .133)
  .5(.5/g)02
• y .0266 ft or ytotal .1333 .0266 .1599 ft

21

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable.

22

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable.

-xA
-xB
23

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB

-xA
-xB
24

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2

-xA
-xB
25

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB

-xA
-xB
26

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB

-xA
-xB
x
F
A
NA
10g
27

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2

-xA
-xB
x
F
A
NA
10g
28

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5)

-xA
-xB
x
F
A
NA
10g
29

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22

-xA
-xB
x
F
A
NA
10g
30

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
31

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B T1 U12 T2
• (1/2)8(0)2

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
32

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B T1 U12 T2
• (1/2)8(0)2 (3F 8gsin30)( DxB)

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
33

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B T1 U12 T2
• (1/2)8(0)2 (3F 8gsin30)( DxB) (1/2)8vB22

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
34

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B T1 U12 T2
• (1/2)8(0)2 (3F 8gsin30)( DxB) (1/2)8vB22
• 4 equations and 4 unknowns DxB, F, vA2, and vB2

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
35

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B T1 U12 T2
• (1/2)8(0)2 (3F 8gsin30)( DxB) (1/2)8vB22
• 4 equations and 4 unknowns DxB, F, vA2, and vB2
• (F 5g)(-.5) (1/2)10vA22
• (3F 4g)(.5/3) (1/2)(8/9)vA22

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
36

• Problems involving dependent motion
• Problem 13.21 The two blocks shown are released
• from rest. Neglecting the masses of the pulleys
  and
• the effect of friction in the pulleys and between
  the
• blocks and the incline, determine (a) the
  velocity of
• block A after it has moved 0.5 m, (b) the tension
  in
• the cable. L -xA 3xB If state 1 is the first
  position
• and state 2 is after block A moves .5 m then
• L - xA1 3xB1 and L - xA2 3xB2?0 -DxA3
  DxB
• 0 - vA - 3vB
• For Block A T1 U12 T2
• (1/2)10(0)2 (F 10gsin30)(-.5) (1/2)10vA22
• For Block B T1 U12 T2
• (1/2)8(0)2 (3F 8gsin30)( DxB) (1/2)8vB22
• 4 equations and 4 unknowns DxB, F, vA2, and vB2
• (F 5g)(-.5) (1/2)10vA22
• (3F 4g)(.5/3) (1/2)(8/9)vA22
• vA2 1.82 m/s and F 16.02 N

-xA
-xB
x
F
A
NA
10g
3F
x
NB
8g
37

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.

38
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.

39
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN 0.4mgcos15

40
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off a B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN (0.4)mgcos15
• 
  TA UAB TB

41
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN (0.4)mgcos15
• 
  TA UAB TB
• (1/2)mv02

42
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN (0.4)mgcos15
• 
  TA UAB TB
• (1/2)mv02 (mgsin15 (0.4)mgcos15)6

43
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN (0.4)mgcos15
• 
  TA UAB TB
• (1/2)mv02 (mgsin15
  (0.4)mgcos15)6 (1/2)m22

44
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN (0.4)mgcos15
• 
  TA UAB TB
• (1/2)mv02 (mgsin15
  (0.4)mgcos15)6 (1/2)m22
• v02 (sin15
  (0.4)cos15)12g 4
• 
  

45
x
y
mkN
N
mg

• Problems with kinetic friction (sliding)
• Problem 13.11 Boxes are transported by a conveyor
  belt with a velocity v0
• to a fixed incline at A where they slide and
  eventually fall off at B. Knowing
• that mk 0.40, determine the velocity of the
  conveyor belt if the boxes leave the
• incline at B with a velocity of 2 m/s.
• SFY maY ? N mgcos15 m0
• N mgcos15 ? mkN (0.4)mgcos15
• 
  TA UAB TB
• (1/2)mv02 (mgsin15
  (0.4)mgcos15)6 (1/2)m22
• v02 (sin15
  (0.4)cos15)12g 4
• 
  v0 4.36 m/s