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