The Conical Pendulum
The Conical PendulumA
P
The Conical PendulumA
h
rO P
The Conical PendulumA Force Diagram
h
rO P
The Conical PendulumA Force Diagram
h
rO P
The Conical PendulumA Force Diagram
h
rO P
TT= tension in the string
(always away from object)
The Conical PendulumA
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
The Conical PendulumA
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
The Conical PendulumA
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
The Conical PendulumA
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
The Conical PendulumA
rmvxm
2
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal 0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
sinT
0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
sinT
rmvT
2
sin
0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
sinT
rmvT
2
sin 2mr
0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
sinT
rmvT
2
sin 2mr
0forcesvertical
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
sinT
rmvT
2
sin 2mr
0forcesvertical
mgcosT
The Conical PendulumA
rmvxm
2
0ym
mg
Force Diagram
h
rO P
TT= tension in the string
(always away from object)
icalwith vertmakesstring
pendulumoflocity angular ve
Resultant Forces
rmv2
forces horizontal
sinT
rmvT
2
sin 2mr
0forcesvertical
mgcosT
mgTmgT
cos0cos
mgrmv
TT 1
cossin 2
mgrmv
TT 1
cossin 2
rgv2
tan
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
hr
rgv
2
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
hr
rgv
2
2
2
vgrh
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
hr
rgv
2
2
2
vgrh
2
g
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
hr
rgv
2
2
2
vgrh
2
g
Implications
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
hr
rgv
2
2
2
vgrh
2
g
Implications
•depth of the pendulum below A is independent of the length of the string.
mgrmv
TT 1
cossin 2
rgv2
tan
gr 2
hrAOP tan in But
hr
rgv
2
2
2
vgrh
2
g
Implications
•depth of the pendulum below A is independent of the length of the string.
•as the speed increases, the particle (bob) rises.
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
T
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
T
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal 0forcesvertical
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal 0forcesvertical
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal
sinT
0forcesvertical
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal
sinT2sin mrT
0forcesvertical
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal
sinT2sin mrT
0forcesvertical
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal
sinT2sin mrT
0forcesvertical
mg
cosT
e.g. The number of revolutions per minute of a conical pendulum increases from 60 to 90.Find the rise in the level of the bob.
mg
Th
r
rmv2
forces horizontal
sinT2sin mrT
0forcesvertical
mg
cosT
mgTmgT
cos0cos
gr
mgmr
2
2
1tan
gr
mgmr
2
2
1tan
hr
tanBut
gr
mgmr
2
2
1tan
hr
tanBut
2
2
gh
hr
gr
gr
mgmr
2
2
1tan
hr
tanBut
2
2
gh
hr
gr
rad/s2
rad/s60
12060rev/minwhen
gr
mgmr
2
2
1tan
hr
tanBut
2
2
gh
hr
gr
rad/s2
rad/s60
12060rev/minwhen
m4
2
2
2
g
gh
gr
mgmr
2
2
1tan
hr
tanBut
2
2
gh
hr
gr
rad/s2
rad/s60
12060rev/minwhen
m4
2
2
2
g
gh
rad/s3
rad/s60
180rev/min09when
gr
mgmr
2
2
1tan
hr
tanBut
2
2
gh
hr
gr
rad/s2
rad/s60
12060rev/minwhen
m4
2
2
2
g
gh
rad/s3
rad/s60
180rev/min09when
m9
3
2
2
g
gh
gr
mgmr
2
2
1tan
hr
tanBut
2
2
gh
hr
gr
rad/s2
rad/s60
12060rev/minwhen
m4
2
2
2
g
gh
rad/s3
rad/s60
180rev/min09when
m9
3
2
2
g
gh
0.14m
m94
heightin rise 22
gg
(ii) (2002)A particle of mass m is suspended by a string of length l from a point directly above the vertex of a smooth cone, which has a vertical axis. The particle remains in contact with the cone and rotates as a conical pendulum with angular velocity .
The angle of the cone at its vertex iswhere , and the string makes an
2
4
angle of with the horizontal as shown in the diagram. The forces acting on the particle are the tension in the string T, the normal reaction N and the gravitational force mg.
(ii) (2002)A particle of mass m is suspended by a string of length l from a point directly above the vertex of a smooth cone, which has a vertical axis. The particle remains in contact with the cone and rotates as a conical pendulum with angular velocity .
The angle of the cone at its vertex iswhere , and the string makes an
2
4
angle of with the horizontal as shown in the diagram. The forces acting on the particle are the tension in the string T, the normal reaction N and the gravitational force mg.
Note: whenever a particle makes contact with a surface there will be a normal force perpendicular to the surface.
a) Show, with the aid of a diagram, that the vertical component of N is sinN
a) Show, with the aid of a diagram, that the vertical component of N is sinN
a) Show, with the aid of a diagram, that the vertical component of N is sinN
T
a) Show, with the aid of a diagram, that the vertical component of N is sinN
T
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
cosT cosN
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
cosT2coscos mrNT
cosN
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
cosT2coscos mrNT
cosN
0forcesvertical
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
cosT2coscos mrNT
cosN
0forcesvertical
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
cosT2coscos mrNT
cosN
0forcesvertical
mgsinT sinN
a) Show, with the aid of a diagram, that the vertical component of N is sinN
NT
mg
c
sin
sin
NcNc
sinisof component verticalthe
NN
and , of terms
in for expressionan find and ,sin
that Show b)
lm
NTmgNT
2forces horizontal mr
cosT2coscos mrNT
cosN
0forcesvertical
mgsinT
mgNTmgNT
sinsin0sinsin
sinN
mgNT sinsin
mgNT sinsin
sin
sinmgNT
mgNT
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
and , of in terms of valuefor this expressionan Find cone. th thecontact wi lose about to is
particlewhen theis,that 0,untilincreasedislocity angular veThe c)
gl
N
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
and , of in terms of valuefor this expressionan Find cone. th thecontact wi lose about to is
particlewhen theis,that 0,untilincreasedislocity angular veThe c)
gl
N
;0When N
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
and , of in terms of valuefor this expressionan Find cone. th thecontact wi lose about to is
particlewhen theis,that 0,untilincreasedislocity angular veThe c)
gl
N
;0When N
sinmgT
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
and , of in terms of valuefor this expressionan Find cone. th thecontact wi lose about to is
particlewhen theis,that 0,untilincreasedislocity angular veThe c)
gl
N
;0When N
sinmgT 2 and mlT
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
and , of in terms of valuefor this expressionan Find cone. th thecontact wi lose about to is
particlewhen theis,that 0,untilincreasedislocity angular veThe c)
gl
N
;0When N
sinmgT 2 and mlT
2
sin
mlmg
mgNT sinsin
sin
sinmgNT
mgNT
2coscos mrNT
cos
cos2
2
mrNT
mrNT
cosBut lr 2mlNT
and , of in terms of valuefor this expressionan Find cone. th thecontact wi lose about to is
particlewhen theis,that 0,untilincreasedislocity angular veThe c)
gl
N
;0When N
sinmgT 2 and mlT
2
sin
mlmg
sin
sin2
lg
lg
Exercise 9C; all
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