Final Exam Thursday December 15th in A111

1

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

2

• Application Conservation of energy to problems
  with vibrations
• Step 1 Put the datum line through the center of
  mass in the static equilibrium position

3

• Application Conservation of energy to problems
  with vibrations
• Step 1 Put the datum line through the center of
  mass in the static equilibrium position
• Step 2 Displace the system to its maximum
  displacement and call this state one

4

• Application Conservation of energy to problems
  with vibrations
• Step 1 Put the datum line through the center of
  mass in the static equilibrium position
• Step 2 Displace the system to its maximum
  displacement and call this state one
• Step 3 At this point the velocity is zero so T1
  0, Calculate V1 and the sum of all elastic and
  gravity potential energies

5

• Application Conservation of energy to problems
  with vibrations
• Step 1 Put the datum line through the center of
  mass in the static equilibrium position
• Step 2 Displace the system to its maximum
  displacement and call this state one
• Step 3 At this point the velocity is zero so T1
  0, Calculate V1 and the sum of all elastic and
  gravity potential energies
• Step 4 At the equilibrium position, state two,
  calculate both T2 and V2

6

• Application Conservation of energy to problems
  with vibrations
• Step 1 Put the datum line through the center of
  mass in the static equilibrium position
• Step 2 Displace the system to its maximum
  displacement and call this state one
• Step 3 At this point the velocity is zero so T1
  0, Calculate V1 and the sum of all elastic and
  gravity potential energies
• Step 4 At the equilibrium position, state two,
  calculate both T2 and V2
• Step 5 Substitute either v xMwN or w qMwN and
  solve for wN

7

• Application Conservation of energy to problems
  with vibrations
• Step 1 Put the datum line through the center of
  mass in the static equilibrium position
• Step 2 Displace the system to its maximum
  displacement and call this state one
• Step 3 At this point the velocity is zero so T1
  0, Calculate V1 and the sum of all elastic and
  gravity potential energies
• Step 4 At the equilibrium position, state two,
  calculate both T2 and V2
• Step 5 Substitute either v xMwN or w qMwN and
  solve for wN
• Step 6 Solve for any unknown Quantities

8

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.

BY
FE
BX
qM
mg
mg
9

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM

BY
FE
BX
qM
mg
mg
10

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM

BY
FE
BX
qM
mg
mg
11

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0

BY
FE
BX
qM
mg
mg
12

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL

BY
FE
BX
qM
mg
mg
13

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2

BY
FE
BX
qM
mg
mg
14

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2

BY
FE
BX
qM
mg
mg
15

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2 and T2 (1/2)(m(L/2)2
  mL2)w2

BY
FE
BX
qM
mg
mg
16

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2 and T2 (1/2)(m(L/2)2
  mL2)w2
• T1 V1 T2 V2
• 0 (1/2)kdST2(1/8)kL2qM2

BY
FE
BX
qM
mg
mg
17

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2 and T2 (1/2)(m(L/2)2
  mL2)w2
• T1 V1 T2 V2
• 0 (1/2)kdST2(1/8)kL2qM2(1/2)kdST2(5/8)mL2w2

BY
FE
BX
qM
mg
mg
18

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2 and T2 (1/2)(m(L/2)2
  mL2)w2
• T1 V1 T2 V2
• 0 (1/2)kdST2(1/8)kL2qM2 (1/2)kdST2(5/8)mL2w2
• (1/8)kL2qM2 (5/8)mL2 wN2qM2

BY
FE
BX
qM
mg
mg
19

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2 and T2 (1/2)(m(L/2)2
  mL2)w2
• T1 V1 T2 V2
• 0 (1/2)kdST2(1/8)kL2qM2(1/2)kdST2(5/8)mL2w2
• (1/8)kL2qM2 (5/8)mL2 wN2qM2

BY
FE
BX
qM
mg
mg
20

• Problem 19.69 Two small spheres, A and C, each
• of mass M, are attached to rod AB, which is
• supported by a pin and bracket at B and a spring
• CD of constant k. Knowing that the mass of the
• rod is negligible and that the system is in
• equilibrium when the rod is horizontal, determine
• frequency of the small oscillations of the
  system.
• V1 (1/2)k(dST (L/2)qM)2 mg(L/2)qM mgLqM
• V1 (1/2)k(dST (L/2)qM)2 (3/2)mgLqM
• V1(1/2)kdST2 (1/2)kdSTLqM(1/8)kL2qM2
• 
  (3/2)mgLqM
• At equilibrium sum moments at B
• mgL mgL/2 kdST(L/2) 0
• kdST(L/2) (3/2)mgL
• V1(1/2)kdST2 (1/8)kL2qM2
• V2(1/2)kdST2 and T2 (1/2)(m(L/2)2
  mL2)w2
• T1 V1 T2 V2
• 0 (1/2)kdST2(1/8)kL2qM2(1/2)kdST2(5/8)mL2w2
• (1/8)kL2qM2 (5/8)mL2 wN2qM2

BY
FE
BX
qM
mg
mg
21

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.

OY
OX
q
FS
22

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0

State 2
OY
OX
qM
FS
State 1
23

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0
• T2 (1/2)(3/g)(1.5wM)2 (1/2)(3/g)(2wM)2

State 2
OY
OX
qM
FS
State 1
24

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0
• T2 (1/2)(3/g)(1.5wM)2 (1/2)(3/g)(2wM)2
• T2 (75/8g)wM2
• T1 V1 T2 V2

State 2
OY
OX
qM
FS
State 1
25

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0
• T2 (1/2)(3/g)(1.5wM)2 (1/2)(3/g)(2wM)2
• T2 (75/8g)wM2
• T1 V1 T2 V2
• 0 2(48)qM2 (75/8g)wM2 0

State 2
OY
OX
qM
FS
State 1
26

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0
• T2 (1/2)(3/g)(1.5wM)2 (1/2)(3/g)(2wM)2
• T2 (75/8g)wM2
• T1 V1 T2 V2
• 0 2(48)qM2 (75/8g)wM2 0
• wM qMwN ? 2(48)qM2 (75/8g)qM2wN2

State 2
OY
OX
qM
FS
State 1
27

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0
• T2 (1/2)(3/g)(1.5wM)2 (1/2)(3/g)(2wM)2
• T2 (75/8g)wM2
• T1 V1 T2 V2
• 0 2(48)qM2 (75/8g)wM2 0
• wM qMwN ? 2(48)qM2 (75/8g)qM2wN2
• wN 18.16 rad/s

State 2
OY
OX
qM
FS
State 1
28

• Problem 19.72 Two blocks, each of weight 3 lb,
  are
• attached to links which are pin connected to bar
  BC
• as shown. The weights of the links and bar are
• negligible, and the blocks can slide without
  friction.
• Block D is attached to a spring of constant k
  4lb/in.
• Knowing that block A is at rest when it is struck
• horizontally with a mallet and given an initial
  velocity
• of 10 in/s, determine the magnitude of the
  maximum
• displacement of block D during the resulting
  motion.
• V1 (1/2)k(2qM)2 V2 0
• T2 (1/2)(3/g)(1.5wM)2 (1/2)(3/g)(2wM)2
• T2 (75/8g)wM2
• T1 V1 T2 V2
• 0 2(48)qM2 (75/8g)wM2 0
• wM qMwN ? 2(48)qM2 (75/8g)qM2wN2
• wN 18.16 rad/s
• wM qMwN
• (10/12) qM18.16(1.5) ?qM .03059 rad
• xDMAX (24/12).03059 .06119 ft or .734 in

State 2
OY
OX
qM
FS
State 1
29

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.

30

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
31

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
32

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
33

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
34

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2
• In the equilibrium position SMP IPa
• mg.1sin14 kdst.1 IP0 ? mg.1sin14 kdst.1

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
35

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2
• In the equilibrium position SMP IPa
• mg.1sin14 kdst.1 IP0 ? mg.1sin14 kdst.1
• T1 V1 T2 V2
• 0 - kdst.1qM (1/2)kdst2 kdst.1qM
  (1/2)k(.1qM)2
• 
  (3/4)7(.1)2w2 (1/2)kdst2

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
36

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2
• In the equilibrium position SMP IPa
• mg.1sin14 kdst.1 IP0 ? mg.1sin14 kdst.1
• T1 V1 T2 V2
• 0 - kdst.1qM (1/2)kdst2 kdst.1qM
  (1/2)k(.1qM)2
• 
  (3/4)7(.1)2w2 (1/2)kdst2
• (1/2)k(.1qM)2 (3/4)7(.1)2w2

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
37

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2
• In the equilibrium position SMP IPa
• mg.1sin14 kdst.1 IP0 ? mg.1sin14 kdst.1
• T1 V1 T2 V2
• 0 - kdst.1qM (1/2)kdst2 kdst.1qM
  (1/2)k(.1qM)2
• 
  (3/4)7(.1)2w2 (1/2)kdst2
• (1/2)k(.1qM)2 (3/4)7(.1)2w2 and wM qMwN
• wN (2k/3)1/2 (2(800)/3)1/2 8.73 rad/s

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
38

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2
• In the equilibrium position SMP IPa
• mg.1sin14 kdst.1 IP0 ? mg.1sin14 kdst.1
• T1 V1 T2 V2
• 0 - kdst.1qM (1/2)kdst2 kdst.1qM
  (1/2)k(.1qM)2
• 
  (3/4)7(.1)2w2 (1/2)kdst2
• (1/2)k(.1qM)2 (3/4)7(.1)2w2 and wM qMwN
• wN (2k/3)1/2 (2(800)/3)1/2 8.73 rad/s
• tN 2p/wN 2p/8.73 .719 s

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg
39

• Problem 19.78 A 7 kg uniform cylinder can roll
• without sliding on an incline and is attached to
  a
• spring AB as shown. If the center of the cylinder
  is
• moved .01 m down the incline and released,
• determine (a) the period of vibration, (b) the
• maximum velocity of the center of the cylinder.
• V1 - mg.1qMsin14 (1/2)k(dst .1qM)2
• V2 (1/2)kdst2
• T2 (1/2)((1/2)mr2 mr2)w2 (3/4)7(.1)2w2
• In the equilibrium position SMP IPa
• mg.1sin14 kdst.1 IP0 ? mg.1sin14 kdst.1
• T1 V1 T2 V2
• 0 - kdst.1qM (1/2)kdst2 kdst.1qM
  (1/2)k(.1qM)2
• 
  (3/4)7(.1)2w2 (1/2)kdst2
• (1/2)k(.1qM)2 (3/4)7(.1)2w2 and wM qMwN
• wN (2k/3)1/2 (2(800)/3)1/2 8.73 rad/s
• tN 2p/wN 2p/8.73 .719 s
• vMAX .1qMwN .1(.01)(8.73)/(.1) .0873 m/s

State 2
State 1
.01m
F2
Datum
F1
f
N
f
N
mg
mg