Final Exam Thursday December 15th in A111

1

• Final Exam Thursday December 15th in A111
• 1010 to 1210
• The final is comprehensive

2

• For simple harmonic motion the general solution
  is
• x C1sin(k/m)1/2t C2 cos(k/m)1/2t
• C1 and C2 are constants determined by constraints
  of the problem
• The natural or circular frequency wN (k/m)1/2
• x C1sinwN t C2 coswN t
• We can introduce a phase angle f
• x xMAXsin(wNt f)

3

• For simple harmonic motion the general solution
  is
• x C1sin(k/m)1/2t C2 cos(k/m)1/2t
• C1 and C2 are constants determined by constraints
  of the problem
• The natural or circular frequency wN (k/m)1/2
• x C1sinwN t C2 coswNt
• We can introduce a phase angle f
• x xMAXsin(wNt f)
• v xMAXwNsin(wNt f)
• a - xMAXwN2sin(wNt f)

4

• For simple harmonic motion the general solution
  is
• x C1sin(k/m)1/2t C2 cos(k/m)1/2t
• C1 and C2 are constants determined by constraints
  of the problem
• The natural or circular frequency wN (k/m)1/2
• x C1sinwN t C2 coswNt
• We can introduce a phase angle f
• x xMAXsin(wNt f)
• v xMAXwNsin(wNt f)
• a - xMAXwN2sin(wNt f)
• The unknown constant is now f
• f is the distance the sin curve is offset to the
• left

5

• Systems of springs
• 2 springs in parallel k1 and k2
• k k1 k2

6

• Systems of springs
• 2 springs in parallel k1 and k2
• k k1 k2
• 2 springs in series k1 and k2
• k k1k2/(k1 k2)

7

• Systems of springs
• 2 springs in parallel k1 and k2
• k k1 k2
• 2 springs in series k1 and k2
• k k1k2/(k1 k2)
• More than 2 springs reduce to 2 by finding
  equivalent k

8

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k

A
A
C
B
9

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m

A
A
C
B
10

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m

A
A
C
B
11

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m
• wN (k/m)1/2 (4110/13.6)1/2 17.39

A
A
C
B
12

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m
• wN (k/m)1/2 (4110/13.6)1/2 17.39
• tN 2p/wN 2p/17.39 .361 s

A
A
C
B
13

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m
• wN (k/m)1/2 (4110/13.6)1/2 17.39
• tN 2p/wN 2p/17.39 .361 s
• fN 1/tN 1/.361 2.77 Hz

A
A
C
B
14

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m
• wN (k/m)1/2 (4110/13.6)1/2 17.39
• tN 2p/wN 2p/17.39 .361 s
• fN 1/tN 1/.361 2.77 Hz
• vMAX xmaxwN .044(17.39) .765 m/s

A
A
C
B
15

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m
• wN (k/m)1/2 (4110/13.6)1/2 17.39
• tN 2p/wN 2p/17.39 .361 s
• fN 1/tN 1/.361 2.77 Hz
• vMAX xmaxwN .044(17.39) .765 m/s
• aMAX xmaxwN2 .044(17.39)2 13.31 m/s2

A
A
C
B
16

• Problem 19.20 A 13.6 kg block is supported by the
  3
• springs shown. If the block is moved from its
• equilibrium .044 m vertically downward and
  released,
• determine (a) the period and the frequency of the
• resulting motion, (b) the maximum velocity and
• acceleration of the block.
• Find the equivalent k
• Springs A and B are in series
• k 3.5(2.1)/(3.5 2.1) 1.31 kN/m
• Springs A and B are in parallel with spring C
• k 1.31 2.8 4.11 kN/m
• wN (k/m)1/2 (4110/13.6)1/2 17.39
• tN 2p/wN 2p/17.39 .361 s
• fN 1/tN 1/.361 2.77 Hz
• vMAX xmaxwN .044(17.39) .765 m/s
• aMAX xmaxwN2 .044(17.39)2 13.31 m/s2
• x xMAXsin(wNt f) .044sin(17.39t f)
• At t 0, x .044
• .044 .044sin(17.39(0) f) ? 1 sinf ? f p/2

A
A
C
B
17

• Simple pendulum

18

• Simple pendulum
• SFT maT
• - mgsinq mLa

19

• Simple pendulum
• SFT maT
• - mgsinq mLa
• d2q/dt2 (g/L)sinq 0
• For small angles q sinq
• d2q/dt2 (g/L)q 0

20

• Simple pendulum
• SFT maT
• - mgsinq mLa
• d2q/dt2 (g/L)sinq 0
• For small angles q sinq
• d2q/dt2 (g/L)q 0
• q qMAXsin(wN f) where wN (g/L)1/2

21
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A

q
mg
B
22
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A
• SMA IAa
• - (L/2)mgsinq (1/3)mL2a

q
mg
B
23
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A
• SMA IAa
• - (L/2)mgsinq (1/3)mL2a
• d2q/dt2 (3g/2L)sinq 0
• d2q/dt2 (3g/2L)q 0
• q qMAXsin(wN f) where wN (3g/2L)1/2

q
mg
B
24
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A
• SMA IAa
• - (L/2)mgsinq (1/3)mL2a
• d2q/dt2 (3g/2L)sinq 0
• d2q/dt2 (3g/2L)q 0
• q qMAXsin(wN f) where wN (3g/2L)1/2
• Step 1 perturb the rigid body from equilibrium
• and draw FBD

q
mg
B
25
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A
• SMA IAa
• - (L/2)mgsinq (1/3)mL2a
• d2q/dt2 (3g/2L)sinq 0
• d2q/dt2 (3g/2L)q 0
• q qMAXsin(wN f) where wN (3g/2L)1/2
• Step 1 perturb the rigid body from equilibrium
• and draw FBD
• Step 2 Sum moments about a fixed point

q
mg
B
26
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A
• SMA IAa
• - (L/2)mgsinq (1/3)mL2a
• d2q/dt2 (3g/2L)sinq 0
• d2q/dt2 (3g/2L)q 0
• q qMAXsin(wN f) where wN (3g/2L)1/2
• Step 1 perturb the rigid body from equilibrium
• and draw FBD
• Step 2 Sum moments about a fixed point
• Step 3 Put the resulting equation in standard
• form d2q/dt2 Kq 0

q
mg
B
27
A

• Free vibrations of rigid bodies
• Consider a thin bar AB of length L and mass m
• Supported by a hinge at A
• SMA IAa
• - (L/2)mgsinq (1/3)mL2a
• d2q/dt2 (3g/2L)sinq 0
• d2q/dt2 (3g/2L)q 0
• q qMAXsin(wN f) where wN (3g/2L)1/2
• Step 1 perturb the rigid body from equilibrium
• and draw FBD
• Step 2 Sum moments about a fixed point
• Step 3 Put the resulting equation in standard
• form d2q/dt2 Kq 0
• Step 4 Generate the solution and solve for the
• unknown quantities

q
mg
B
28

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.

FB
CY
q
CX
Y
FA
X
29

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq

FB
CY
q
CX
Y
FA
X
30

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a

FB
CY
q
CX
Y
FA
X
31

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a
• a 526.19sinq 0
• For small angles a 526.19q 0

FB
CY
q
CX
Y
FA
X
32

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a
• a 526.19sinq 0
• For small angles a 526.19q 0
• q qMAXsin(wN f) where wN 526.191/2

FB
CY
q
CX
Y
FA
X
33

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a
• a 526.19sinq 0
• For small angles a 526.19q 0
• q qMAXsin(wN f) where wN 526.191/2
• q qMAXsin(22.94 f)

FB
CY
q
CX
Y
FA
X
34

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a
• a 526.19sinq 0
• For small angles a 526.19q 0
• q qMAXsin(wN f) where wN 526.191/2
• q qMAXsin(22.94 f)
• fN wN/2p 22.94/2p 3.65 Hz

FB
CY
q
CX
Y
FA
X
35

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a
• a 526.19sinq 0
• For small angles a 526.19q 0
• q qMAXsin(wN f) where wN 526.191/2
• q qMAXsin(22.94 f)
• fN wN/2p 22.94/2p 3.65 Hz
• vMAX/.36 qMAXwN ? .0011/.36 qMAX22.94

FB
CY
q
CX
Y
FA
X
36

• Problem 19.39 The 9 kg uniform rod AB is
• attached to springs at A and B, each of
• constant 850 N/m, which can act in tension or
• compression. If the end A of the rod is
• depressed slightly and released, determine (a)
• the frequency of the vibration, (b) the
• amplitude of the angular motion of the,
• knowing that the maximum velocity of point A
• is 1.1 mm/s.
• FA k.36sinq FB k.24sinq
• SMC ICa
• -.36k.36sinq -.24k.24sinq ((1/12)9(.6)2
  9(.06)2)a
• a 526.19sinq 0
• For small angles a 526.19q 0
• q qMAXsin(wN f) where wN 526.191/2
• q qMAXsin(22.94 f)
• fN wN/2p 22.94/2p 3.65 Hz
• vMAX/.36 qMAXwN ? .0011/.36 qMAX22.94
• qMAX .000132 rad or .007640

FB
CY
q
CX
Y
FA
X
37

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.

38

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.

CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
39

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX

CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
40

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX
• SMD Ida
• CXr (3/2)(12/g)r2a

D
CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
41

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX -rq d
• SMD Ida
  -ar aABX
• CXr (3/2)(12/g)r2a
• CX - (8/g)ar - 2krq

D
CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
42

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX -rq d
• SMD Ida
  -ar aABX
• CXr (3/2)(12/g)r2a
• CX - (8/g)ar - 2krq
• - (8/g)a - 2kq (3/2)(12/g)a ? a (kg/13)q

D
CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
43

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX -rq d
• SMD Ida
  -ar aABX
• CXr (3/2)(12/g)r2a
• CX - (8/g)ar - 2krq
• - (8/g)a - 2kq (3/2)(12/g)a ? a (kg/13)q
• wN ((30)12(32.2)/13)1/2 29.86

D
CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
44

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX -rq d
• SMD Ida
  -ar aABX
• CXr (3/2)(12/g)r2a
• CX - (8/g)ar - 2krq
• - (8/g)a - 2kq (3/2)(12/g)a ? a (kg/13)q
• wN ((30)12(32.2)/13)1/2 29.86
• t 2p/wN 6.28/29.86 .210 s

D
CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N
45

• 19-42 a 12-lb uniform cylinder can roll without
• sliding on a horizontal surface and is attached
  by
• a pin at point C to an 8 lb horizontal bar AB.
  The
• bar is attached to 2 springs, each of constant k
  
• 30 lb/in. as shown. Knowing that the bar has
• moved .5 in to the right of the equilibrium
  position
• and released, determine (a) the period of
  vibration
• of the system, (b) the magnitude of the maximum
• velocity of bar AB.
• SFX maABX
• - 2kd CX (8/g)aABX -rq d
• SMD Ida
  -ar aABX
• CXr (3/2)(12/g)r2a
• CX - (8/g)ar - 2krq
• - (8/g)a - 2kq (3/2)(12/g)a ? a (kg/13)q
• wN ((30)12(32.2)/13)1/2 29.86
• t 2p/wN 6.28/29.86 .210 s
• vmax xmaxwN .5(29.86) 14.93 in/s

D
CY
kd
kd
CX
8
Y
CX
CY
X
f
12
N