SF maG

1

• SF maG
• In rectangular coordinates in the x-y plane
• SFx maGx SFy maGy SMz Izza
• where Izz is the mass moment of inertia about the
  z axis
• Procedure to solve problems
• 1. Draw FBD(s) (Free Body Diagram) showing all
  forces and moments for each rigid body
• 2. Put an axis showing the positive directions on
  each FBD
• 3. Apply the scalar kinetic equations to each FBD
• 4. Generate kinematic equation(s) to eliminate an
  unknown(s)
• 5. Solve the system of equations.

2

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.

3

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.

y
TB
TA
x
G
400
4

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400

y
TB
TA
x
G
400
5

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY

y
TB
TA
x
G
400
6

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3

y
TB
TA
x
G
400
7

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a

y
TB
TA
x
G
400
8

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a

y
TB
TA
x
G
400
9

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A

y
TB
TA
x
G
400
10

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i

y
TB
TA
x
G
400
11

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2

y
TB
TA
x
G
400
12

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2
• aG aA ak x rG/A w2rG/A

y
TB
TA
x
G
400
13

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2
• aG aA ak x rG/A w2rG/A
• aG 3j (1/3.3)k x (3.3i 1.8j)
• 02
  (3.3i 1.8j)

y
TB
TA
x
G
400
14

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2
• aG aA ak x rG/A w2rG/A
• aG 3j (1/3.3)k x (3.3i 1.8j)
• 02
  (3.3i 1.8j)
• aG (1.8/3.3)i 2j

y
TB
TA
x
G
400
15

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2
• aG aA ak x rG/A w2rG/A
• aG 3j (1/3.3)k x (3.3i 1.8j)
• 02
  (3.3i 1.8j)
• aG (1.8/3.3)i 2j
• TA TB 424.84

y
TB
TA
x
G
400
16

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2
• aG aA ak x rG/A w2rG/A
• aG 3j (1/3.3)k x (3.3i 1.8j)
• 02
  (3.3i 1.8j)
• aG (1.8/3.3)i 2j
• TA TB 424.84
• - TA TB - 5.37

y
TB
TA
x
G
400
17

• Problem 16.55 The 400 lb crate shown is
• lowered by means of two overhead cranes.
• Knowing that at the instant shown the
• deceleration of cable A is 3 ft/s2 and that of
• cable B is 1 ft/s2, determine the tension in each
• cable.
• SFy maGy
• TA TB 400 (400/g)aGY
• SMz Izza
• - TA3.3 TB3.3 (1/12)(400/g)(6.62 3.62)a
• - TA TB 17.73a
• aB aA ak x rB/A w2rB/A
• 1j 3j ak x 6.6i 026.6i ?a -1/3.3 rad/s2
• aG aA ak x rG/A w2rG/A
• aG 3j (1/3.3)k x (3.3i 1.8j)
• 02
  (3.3i 1.8j)
• aG (1.8/3.3)i 2j
• TA TB 424.84
• - TA TB - 5.37

y
TB
TA
x
G
400
18

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.

19

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.

FB
FA
G
y
mg
x
20

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy

FB
FA
G
y
mg
x
21

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB - mg m(0)

FB
FA
G
y
mg
x
22

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB - mg m(0)
• SMG Izza

FB
FA
G
y
mg
x
23

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB - mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)

FB
FA
G
y
mg
x
24

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB - mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2

FB
FA
G
y
mg
x
25

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy

FB
FA
G
y
mg
x
26

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy

FB
FA
G
y
mg
x
27

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2

FB
FA
G
y
mg
x
28

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza

FB
FA
G
y
mg
x
29

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a

FB
FA
G
y
mg
x
30

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)

FB
FA
G
y
mg
x
31

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)
• aA aG ak x rA/G w2rA/G

FB
FA
G
y
mg
x
32

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)
• aA aG ak x rA/G w2rA/G
• aA - (g/2)j - (2g/b)k x (- b/2)i 02 (-
  b/2)i

FB
FA
G
y
mg
x
33

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)
• aA aG ak x rA/G w2rA/G
• aA - (g/2)j - (2g/b)k x (- b/2)i 02 (-
  b/2)i
• aA (g/2)j

FB
FA
G
y
mg
x
34

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)
• aA aG ak x rA/G w2rA/G
• aA - (g/2)j - (2g/b)k x (- b/2)i 02 (-
  b/2)i
• aA (g/2)j
• aB aG ak x rB/G w2rB/G

FB
FA
G
y
mg
x
35

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)
• aA aG ak x rA/G w2rA/G
• aA - (g/2)j - (2g/b)k x (- b/2)i 02 (-
  b/2)i
• aA (g/2)j
• aB aG ak x rB/G w2rB/G
• aB - (g/2)j - (2g/b)k x (b/2)i 02 (b/2)i

FB
FA
G
y
mg
x
36

• Problem 16.61 A thin circular plate of mass m is
• suspended from two springs as shown. If spring 2
• breaks, determine the acceleration at that
  instant (a)
• of point A, (b) of point B.
• Before the spring breaks
• SFy maGy ? FA FB mg m(0)
• SMG Izza ? - (b/2)FA (b/2)FB Izz(0)
• FA FB mg/2
• After the spring breaks FB 0
• SFy maGy ? mg/2 - mg maGy ? aGy - g/2
• SMG Izza ? - (b/2)(mg/2) (1/2)mr2a ? a -
  (2g/b)
• aA aG ak x rA/G w2rA/G
• aA - (g/2)j - (2g/b)k x (- b/2)i 02 (-
  b/2)i
• aA (g/2)j
• aB aG ak x rB/G w2rB/G
• aB - (g/2)j - (2g/b)k x (b/2)i 02 (b/2)i
• aB (3g/2)j

FB
FA
G
y
mg
x
37
(No Transcript)
38

• Collar B slides along rod AC and is
• attached to a block that moves in a vertical
• slot. Knowing that R 1.5 ft, q 300,
• w 6 rad/s, and a 4 rad/s2, determine
• the velocity and acceleration of collar B.
• vB ? aB ?

39

• Collar B slides along rod AC and is
• attached to a block that moves in a vertical
• slot. Knowing that R 1.5 ft, q 300,
• w 6 rad/s, and a 4 rad/s2, determine
• the velocity and acceleration of collar B.
• vB ? aB ?
• vB vA wk x rB/A vB/AC
• vBj 0i 6k x (1.5i .866j) vB/AC (.866i
• 
  .5j)
• i ? 0 - 5.20 .866vB/AC
• j ? vB 9 .5vB/AC ? vB 12 ft/s
• vB/AC 6 ft/s
• aB aA ak x rB/A w2 rB/A 2wk x vB/AC
  aB/AC
• aBj 0i 4k x (1.5i .866j ) 62 (1.5i
• .866j) 2(6k x (5.2i 3j )) aB/AC (.866i
  .5j)
• aBj 6j - 3.47i 54i 31.2j 62.4j- 36i
  aB/AC (.866i .5j)
• i ? 0 - 3.47 54 - 36 .866aB/AC
• j ? aB 6 31.2 62.4 .5aB/AC
• ? aB
  91.1 ft/s2

Y
X
40

• Collar B slides along rod AC and is
• attached to a block that moves in a vertical
• slot. Knowing that R 1.5 ft, q 300,
• w 6 rad/s, and a 4 rad/s2, determine
• the velocity and acceleration of collar B.
• vB ? aB ?
• vB vA wk x rB/A vB/AC
• vB(.5i .866j) 0i 6k x 1.73i vB/ACi
• i ? .5vB vB/AC
• j ? .866vB 10.38 ? vB 12 ft/s
• vB/AC 6 ft/s
• aB aA ak x rB/A w2 rB/A 2wk x vB/AC
  aB/AC
• aB (.5i .866j) 0i 4k x 1.73i 621.73i
• 2(6k x 6i) aB/AC i
• .5aBi .866aBj 6.92j 62.3i 72j aB/ACi
• i ? .5aB -62.3 aB/AC
• j ? .866aB 6.92 72
• 
  aB 91.1 ft/s2
• aB/AC
  108 ft/s2

Y
X