vB vA

1

• vB vA AB x rB/A

2

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A

3

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A
• In this problem both A and E have
• 0 velocity since they are pinned
• wAB -4, rB/A 10 in, vB -40 in/s

4

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A
• In this problem both A and E have
• 0 velocity since they are pinned
• wAB -4, rB/A 10 in, vB -40 in/s
• wDE 6.67, rD/E (2.42 62)1/2 in
• vD -43.1 in/s

5

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A
• In this problem both A and E have
• 0 velocity since they are pinned
• wAB -4, rB/A 10 in, vB -40 in/s
• wDE 6.67, rD/E (2.42 62)1/2 in
• vD -43.1 in/s

6

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A
• In this problem both A and E have
• 0 velocity since they are pinned
• wAB -4, rB/A 10 in, vB -40 in/s
• wDE 6.67, rD/E (2.42 62)1/2 in
• vD -43.1 in/s

7

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A
• In this problem both A and E have
• 0 velocity since they are pinned
• wAB -4, rB/A 10 in, vB -40 in/s
• wDE 6.67, rD/E (2.42 62)1/2 in
• vD -43.1 in/s
• wBD 40/10 4 rad/s

8

• vB vA AB x rB/A
• If vA 0 then vB/wAB rB/A
• In this problem both A and E have
• 0 velocity since they are pinned
• wAB -4, rB/A 10 in, vB -40 in/s
• wDE 6.67, rD/E (2.42 62)1/2 in
• vD -43.1 in/s
• wBD 40/10 4 rad/s
• Where the lines cross is ICBD
• Instantaneous Center of Zero Velocity
• For link BD

ICBD
9

• Instantaneous Center of Zero Velocity
• Point A on the tire has zero velocity
• if there is no slipping.

10

• Instantaneous Center of Zero Velocity
• Point A on the tire has zero velocity
• if there is no slipping.
• Every point on the tire has a different
• velocity vector.

11

• Instantaneous Center of Zero Velocity
• Point A on the tire has zero velocity
• if there is no slipping.
• Every point on the tire has a different
• velocity vector.
• If we construct a perpendicular to the
• tail of each velocity vector, they all
• cross at the instantaneous center, IC

IC
12

• Instantaneous Center of Zero Velocity
• Point A on the tire has zero velocity
• if there is no slipping.
• Every point on the tire has a different
• velocity vector.
• If we construct a perpendicular to the
• tail of each velocity vector, they all
• cross at the instantaneous center, IC
• If the velocity vectors are parallel add
• a second line connecting the tips of the
• vectors

IC
13

• Instantaneous Center of Zero Velocity
• Point A on the tire has zero velocity
• if there is no slipping.
• Every point on the tire has a different
• velocity vector.
• If we construct a perpendicular to the
• tail of each velocity vector, they all
• cross at the instantaneous center, IC
• If the velocity vectors are parallel add
• a second line connecting the tips of the
• Vectors
• The magnitude of w is the magnitude
• of the v divided by the perpendicular
• distance
• w v/rP/IC

IC
14

• Instantaneous Center of Zero Velocity
• Point A on the tire has zero velocity
• if there is no slipping.
• Every point on the tire has a different
• velocity vector.
• If we construct a perpendicular to the
• tail of each velocity vector, they all
• cross at the instantaneous center, IC
• If the velocity vectors are parallel add
• a second line connecting the tips of the
• Vectors
• The magnitude of w is the magnitude
• of the v divided by the perpendicular
• distance
• w v/rP/IC
• If they cross at infinity w 0

IC
15

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.

16

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.
• We know the magnitude and direction of the
• velocity of end A.
• We know the direction of the velocity of end B.
• Step1 Sketch the velocity vectors for the end
• points

17

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.
• We know the magnitude and direction of the
• velocity of end A.
• We know the direction of the velocity of end B.
• Step1 Sketch the velocity vectors for the end
• points
• Step2 Construct lines perpendicular to the tail
  of
• the velocity vectors

ICAB
18

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.
• We know the magnitude and direction of the
• velocity of end A.
• We know the direction of the velocity of end B.
• Step1 Sketch the velocity vectors for the end
• points
• Step2 Construct lines perpendicular to the tail
  of
• the velocity vectors
• rA/IC .3 .4(.125/.300) .467 m
• Step 3 Calculate the distance from the IC to the
• endpoints

ICAB
19

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.
• We know the magnitude and direction of the
• velocity of end A.
• We know the direction of the velocity of end B.
• Step1 Sketch the velocity vectors for the end
• points
• Step2 Construct lines perpendicular to the tail
  of
• the velocity vectors
• rA/IC .3 .4(.125/.300) .467 m
• Step 3 Calculate the distance from the IC to the
• endpoints
• rB/IC (.1672 .42)1/2 .433 m

ICAB
20

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.
• We know the magnitude and direction of the
• velocity of end A.
• We know the direction of the velocity of end B.
• Step1 Sketch the velocity vectors for the end
• points
• Step2 Construct lines perpendicular to the tail
  of
• the velocity vectors
• rA/IC .3 .4(.125/.300) .467 m
• Step 3 Calculate the distance from the IC to the
• endpoints
• rB/IC (.1672 .42)1/2 .433 m
• wAB 1.4/.467 -3.0 rad/s
• Step 4 Find the unknown(s), either v or w , from
  the

ICAB
21

• Problem 15.27 Rod AB can slide freely along the
• floor and the inclined plane. At the instant
  shown,
• the velocity of end A is 1.4 m/s to the left.
• Determine (a) the angular velocity of the rod,
  (b)
• the velocity of end B of the rod.
• We know the magnitude and direction of the
• velocity of end A.
• We know the direction of the velocity of end B.
• Step1 Sketch the velocity vectors for the end
• points
• Step2 Construct lines perpendicular to the tail
  of
• the velocity vectors
• rA/IC .3 .4(.125/.300) .467 m
• Step 3 Calculate the distance from the IC to the
• endpoints
• rB/IC (.1672 .42)1/2 .433 m
• wAB 1.4/.467 -3.0 rad/s
• Step 4 Find the unknown(s), either v or w , from
  the

ICAB
22

• Step 1 Sketch the velocity vectors for the end
  points
• Step 2 Construct lines perpendicular to the tail
  of the velocity vectors
• If the lines perpendicular to the
  velocity vectors are parallel
• w 0 the body is only translating
• Step 3 Calculate the distance from the IC to the
  endpoints
• Step 4 Find the unknown(s), either v or w , from
  the equation
• vB/wAB rB/A

23

• Step 1 Sketch the velocity vectors for the end
  points
• Step 2 Construct lines perpendicular to the tail
  of the velocity vectors
• If the lines perpendicular to the
  velocity vectors are parallel
• w 0 the body is only translating
• If the perpendicular to the velocity
  vectors are the same line
• add another line connecting the tips
  of the velocity vectors
• Step 3 Calculate the distance from the IC to the
  endpoints
• Step 4 Find the unknown(s), either v or w , from
  the equation
• vB/wAB rB/A

24

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.

25

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.
• Both the magnitude and the direction of
• the velocity of end A are known.
• The direction of the velocity of end B is
• known.

26

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.
• Both the magnitude and the direction of
• the velocity of end A are known.
• The direction of the velocity of end B is
• known.

ICAB
27

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.
• Both the magnitude and the direction of
• the velocity of end A are known.
• The direction of the velocity of end B is
• known.
• rB/IC/sin65 (20/12)/sin60 ? rB/IC 1.74

ICAB
28

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.
• Both the magnitude and the direction of
• the velocity of end A are known.
• The direction of the velocity of end B is
• known.
• rB/IC/sin65 (20/12)/sin60 ? rB/IC 1.74
• rA/IC/sin55 (20/12)/sin60 ? rA/IC 1.58

ICAB
29

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.
• Both the magnitude and the direction of
• the velocity of end A are known.
• The direction of the velocity of end B is
• known.
• rB/IC/sin65 (20/12)/sin60 ? rB/IC 1.74
• rA/IC/sin55 (20/12)/sin60 ? rA/IC 1.58
• wAB 3.6/1.58 -2.28 rad/s

ICAB
30

• Problem 15.39 Collar A moves up with a
• velocity of 3.6 ft/s. At the instant shown
• when q 250, determine (a) the angular
• velocity of rod AB, (b) the velocity of collar
• B.
• Both the magnitude and the direction of
• the velocity of end A are known.
• The direction of the velocity of end B is
• known.
• rB/IC/sin65 (20/12)/sin60 ? rB/IC 1.74
• rA/IC/sin55 (20/12)/sin60 ? rA/IC 1.58
• wAB 3.6/1.58 -2.28 rad/s
• vB 2.28(1.74) 3.98 ft/s

ICAB
31

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.

IC
32

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.

IC
33

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.

IC
34

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rB/IC .04 m
• rA/IC .10 m

IC
35

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rB/IC .04 m
• rA/IC .10 m
• wAD .12/.04 3 rad/s

IC
36

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rB/IC .04 m
• rA/IC .10 m
• wAD .12/.04 3 rad/s
• vA 3(.10) .30 m/s

IC
37

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rB/IC .04 m
• rA/IC .10 m
• wAD .12/.04 3 rad/s
• vA 3(.10) .30 m/s
• Cable wound .30 - .12 .18 m

IC
38

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.

IC
39

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.

IC
IC
40

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rD/IC .04 m
• rA/IC .06 m

IC
IC
41

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rD/IC .04 m
• rA/IC .06 m
• wAD .12/.04 3 rad/s

IC
IC
42

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rD/IC .04 m
• rA/IC .06 m
• wAD .12/.04 3 rad/s
• vA 3(.06) .18 m/s

IC
IC
43

• Problem 15.75 A 60 mm radius drum is rigidly
• attached to a 100 mm radius drum as shown.
• One of the drums rolls without sliding on the
• surface shown, and a cord is wound around the
• other drum. Knowing that end E of the cord is
• pulled to the left with a velocity of 120 mm/s,
• determine (a) the angular velocity of the drums,
• (b) the velocity of the center of the drums, (c)
  the
• length of cord wound or unwound per second.
• rD/IC .04 m
• rA/IC .06 m
• wAD .12/.04 3 rad/s
• vA 3(.06) .18 m/s
• Cable unwound .18 .12 .3 m

IC
IC
44

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.

45

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.

ICAB
46

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in

ICAB
47

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in
• wAB 6/5.77 1.04 rad/s

ICAB
48

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in
• wAB 6/5.77 1.04 rad/s
• vB 1.04(5.77) 6 in/s

ICAB
49

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in
• wAB 6/5.77 1.04 rad/s
• vB 1.04(5.77) 6 in/s

ICAB
ICBD
50

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in
• wAB 6/5.77 1.04 rad/s
• vB 1.04(5.77) 6 in/s
• rB/IC 2(10cos30) 17.32 in
• rD/IC 10 in

ICAB
ICBD
51

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in
• wAB 6/5.77 1.04 rad/s
• vB 1.04(5.77) 6 in/s
• ICBDB 2(10cos30) 17.32 in
• ICBDD 10 in
• wBD 6/17.32 -.346 rad/s

ICAB
ICBD
52

• Problem 15.91 Two rods AB and BD are
• connected to three collars as shown.
• Knowing that collar A moves downward
• with a velocity of 6 in/s, determine at the
• instant shown (a) the angular velocity of
• each rod, (b) the velocity of collar D.
• rA/IC rB/IC 5/cos30 5.77 in
• wAB 6/5.77 1.04 rad/s
• vB 1.04(5.77) 6 in/s
• ICBDB 2(10cos30) 17.32 in
• ICBDD 10 in
• wBD 6/17.32 -.346 rad/s
• vD .346(10) 3.46 in/s

ICAB
ICBD