Problem 13'69 A 500 g collar can slide without friction

1

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.

2

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2

3

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0

4

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0

y
5

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15

y
6

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2

y
7

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm

y
8

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm

y
9

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm
• T2 (1/2).5v2 .25v2

y
10

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm
• T2 (1/2).5v2 .25v2
• 0 .5(9.81).15 22.99 .25v2 17.48

y
11

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm
• T2 (1/2).5v2 .25v2
• 0 .5(9.81).15 22.99 .25v2 17.48
• v 5 m/s

y
12

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm
• T2 (1/2).5v2 .25v2
• 0 .5(9.81).15 22.99 .25v2 17.48
• v 5 m/s

y
eN
z
FSX
NZ
Ny
.5g
13

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm
• T2 (1/2).5v2 .25v2
• 0 .5(9.81).15 22.99 .25v2 17.48
• v 5 m/s

y
eN
z
FSX
NZ
Ny
.5g
14

• Problem 13.69 A 500 g collar can slide without
  friction
• along a semicircular rod BCD. The spring is of
• constant 320 N/m and its undeformed length is
• 200mm. Knowing that the collar is released from
  rest
• at B, determine (a) the speed of the collar as it
• passes through C, (b) the force exerted by the
  rod on
• the collar at C.
• No friction so use conservation of energy
• T1 V1 T2 V2
• Starts from rest T1 0
• Put datum on z axis so V2G 0
• V1G mgy .5(9.81).15
• V1E .5k(AB -.2)2 .5(320)((.32 .152
  .0752)1/2 - .2)2
• V1E 22.99 Nm
• V2E 5k(AC -.2)2 .5(320)((.32 02 .0752)1/2
  - .2)2
• V2E 17.48 Nm
• T2 (1/2).5v2 .25v2
• 0 .5(9.81).15 22.99 .25v2 17.48
• v 5 m/s

y
eN
z
FSX
NZ
Ny
.5g
15

• Problem 13.93 2 identical 2 kg collars, A
• and B are attached by a spring of
• constant 100 N/m and can slide on a
• horizontal rod which is free to rotate
• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.

16

• Problem 13.93 2 identical 2 kg collars, A
• and B are attached by a spring of
• constant 100 N/m and can slide on a
• horizontal rod which is free to rotate
• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25Nm

17

• Problem 13.93 2 identical 2 kg collars, A
• and B are attached by a spring of
• constant 100 N/m and can slide on a
• horizontal rod which is free to rotate
• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5

er
kx
A
18

• Problem 13.93 2 identical 2 kg collars, A
• and B are attached by a spring of
• constant 100 N/m and can slide on a
• horizontal rod which is free to rotate
• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25 Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5
• V1E .5(100)(.5)2 12.5 Nm

er
kx
A
19

• Problem 13.93 2 identical 2 kg collars, A
• and B are attached by a spring of
• constant 100 N/m and can slide on a
• horizontal rod which is free to rotate
• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25 Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5
• V1E .5(100)(.5)2 12.5 Nm
• Conservation of angular Momentum
• B ? (rmv)1 (rmv)2

er
kx
A
20

• constant 100 N/m and can slide on a
• horizontal rod which is free to rotate
• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25 Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5
• V1E .5(100)(.5)2 12.5 Nm
• Conservation of angular Momentum
• B ? (rmv)1 (rmv)2
• 2.5(2)2.5(5) 3(2)3(dq/dt)2
• (dq/dt)2 3.47 rad/s

er
kx
A
21

• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25 Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5
• V1E .5(100)(.5)2 12.5 Nm
• Conservation of angular Momentum
• B ? (rmv)1 (rmv)2
• 2.5(2)2.5(5) 3(2)3(dq/dt)2
• (dq/dt)2 3.47 rad/s
• A ? (rmv)1 (rmv)2
• 1(2)1(5) rA22rA23.47 ?rA2 1.2 m

er
kx
A
22

• about a vertical shaft. Collar B is initially
• prevented from sliding by a stop as rod
• rotates at constant rate (dq/dt)0 5 rad/s
• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25 Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5
• V1E .5(100)(.5)2 12.5 Nm
• Conservation of angular Momentum
• B ? (rmv)1 (rmv)2
• 2.5(2)2.5(5) 3(2)3(dq/dt)2
• (dq/dt)2 3.47 rad/s
• A ? (rmv)1 (rmv)2
• 1(2)1(5) rA22rA23.47 ?rA2 1.2 m

er
kx
A
23

• and the spring is in compression with rA
• 1m and rB 2.5m. After the stop is
• removed both collars move out along the
• rod. At the instant when rB 3m,
• determine (a) rA, (b)dq/dt, (c) the total
• kinetic energy. Neglect friction and the
• mass of the rod.
• T1 .5(2)(1(5))2 .5(2)(1(5))2181.25 Nm
• SFr mar? -100x 2(0 1(5)2) ? x .5
• V1E .5(100)(.5)2 12.5 Nm
• Conservation of angular Momentum
• B ? (rmv)1 (rmv)2
• 2.5(2)2.5(5) 3(2)3(dq/dt)2
• (dq/dt)2 3.47 rad/s
• A ? (rmv)1 (rmv)2
• 1(2)1(5) rA22rA23.47 ?rA2 1.2 m
• V2E .5(100)(.2)2 2 Nm
• T1 V1 T2 V2
• 181.25 12.5 T2 2

er
kx
A
24
(No Transcript)
25

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy

26

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy
• T1 .5m82 32m

27

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy
• T1 .5m82 32m
• T2 0

28

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy
• T1 .5m82 32m
• T2 0

x
y
mN
N
mg
29

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy
• T1 .5m82 32m
• T2 0
• SFY maY
• N mgcos30 m0
• N mgcos30
• mN .3mgcos30

x
y
mN
N
mg
30

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy
• T1 .5m82 32m
• T2 0
• SFY maY
• N mgcos30 m0
• N mgcos30
• mN .3mgcos30
• T1 U12 T2
• 32m d(- .3mgcos30 mgsin30) 0

x
y
mN
N
mg
31

• The initial velocity of the block in position
• A is 8 m/s. Knowing the that the
• coefficient of kinetic friction between the
• block and the plane is mk 0.3,
• determine the distance it takes to reach
• position B with zero velocity, if q 300.
• Friction so Use Work Energy
• T1 .5m82 32m
• T2 0
• SFY maY
• N mgcos30 m0
• N mgcos30
• mN .3mgcos30
• T1 U12 T2
• 32m d(- .3mgcos30 mgsin30) 0
• 32 dg(.3(.866) .5)
• d 4.29 m

x
y
mN
N
mg