Simple Harmonic Motion

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

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Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

xx kxx

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

xx kxx

xnx 2

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

xx kxx

xnx 2 (constant needs to be negative)

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

xx

xnx 2

obeys;it then SHM, undergoes particle a If

kxx xnx 2 (constant needs to be negative)

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

xx

xnx 2

obeys;it then SHM, undergoes particle a If

xnvdxd 22

21

kxx xnx 2 (constant needs to be negative)

Simple Harmonic MotionA particle that moves back and forward in such a way that its acceleration at any instant is directly proportional to its distance from a fixed point, is said to undergo Simple Harmonic Motion (SHM)

xx

xnx 2

obeys;it then SHM, undergoes particle a If

xnvdxd 22

21

cxnv

cxnv

222

222

21

21

kxx xnx 2 (constant needs to be negative)

when x = a , v = 0 (a = amplitude)

22

2220 i.e.

anc

can

when x = a , v = 0 (a = amplitude)

22

2220 i.e.

anc

can

22

2222

22222

xanv

xanv

anxnv

when x = a , v = 0 (a = amplitude)

22

2220 i.e.

anc

can

22

2222

22222

xanv

xanv

anxnv

NOTE: 02 v

when x = a , v = 0 (a = amplitude)

22

2220 i.e.

anc

can

22

2222

22222

xanv

xanv

anxnv

NOTE: 02 v

022 xa

when x = a , v = 0 (a = amplitude)

22

2220 i.e.

anc

can

22

2222

22222

xanv

xanv

anxnv

NOTE: 02 v

022 xa

axa

when x = a , v = 0 (a = amplitude)

22

2220 i.e.

anc

can

22

2222

22222

xanv

xanv

anxnv

NOTE: 02 v

022 xa

axa

Particle travels back and forward between x = -a and x = a

when x = a , v = 0 (a = amplitude)

22 xandtdx

22 xandtdx

1-

1-

sin and ve choose ,0

cos and ve- choose ,

;0 when If

x

ax

t

22 xandtdx

1-

1-

sin and ve choose ,0

cos and ve- choose ,

;0 when If

x

ax

t

22

1

xandxdt

22 xandtdx

1-

1-

sin and ve choose ,0

cos and ve- choose ,

;0 when If

x

ax

t

22

1

xandxdt

dxxan

tx

a

22

11

22 xandtdx

1-

1-

sin and ve choose ,0

cos and ve- choose ,

;0 when If

x

ax

t

22

1

xandxdt

dxxan

tx

a

22

11

x

aax

n

1cos

1

22 xandtdx

1-

1-

sin and ve choose ,0

cos and ve- choose ,

;0 when If

x

ax

t

22

1

xandxdt

dxxan

tx

a

22

11

x

aax

n

1cos

1

ax

n

ax

n

1

11

cos1

1coscos1

22 xandtdx

1-

1-

sin and ve choose ,0

cos and ve- choose ,

;0 when If

x

ax

t

22

1

xandxdt

dxxan

tx

a

22

11

x

aax

n

1cos

1

ax

n

ax

n

1

11

cos1

1coscos1

ntax

ntax

ax

nt

cos

cos

cos 1

In general;

In general;

xnx 2

obeys SHM undergoing particleA

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

ntax cos

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

ntax cosamplitude where

sin OR

a

ntax

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

ntax cosamplitude where

sin OR

a

ntax

the particle has;

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

ntax cosamplitude where

sin OR

a

ntax

the particle has;

Tf

nT

1 :frequency

2 :period

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

ntax cosamplitude where

sin OR

a

ntax

the particle has;

Tf

nT

1 :frequency

2 :period

(time for one oscillation)

In general;

xnx 2

obeys SHM undergoing particleA

2222 xanv particle theofpath find tous allows

ntax cosamplitude where

sin OR

a

ntax

the particle has;

Tf

nT

1 :frequency

2 :period

(time for one oscillation)

(number of oscillations per time period)

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

x9

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

x9SHMin moves particle

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

x9SHMin moves particle

32T

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

x9SHMin moves particle

32T

seconds 3

2 ismotion of period

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

x9SHMin moves particle

32T

seconds 3

2 ismotion of period

b) Find the interval in which the particle moves and determine the greatest speed.

e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.

a) Show that P is moving in SHM and state the period of motion.

tx

tx

tx

3sin36

3cos12

3sin4

x9SHMin moves particle

32T

seconds 3

2 ismotion of period

b) Find the interval in which the particle moves and determine the greatest speed.

units/s 12 is speedgreatest theand

44 interval thealong moves particle x

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

cxv

cxv

22

22

4

221

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

cxv

cxv

22

22

4

221

72

346 i.e.

6,3when 22

c

c

vx

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

cxv

cxv

22

22

4

221

72

346 i.e.

6,3when 22

c

c

vx

724 22 xv

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

cxv

cxv

22

22

4

221

72

346 i.e.

6,3when 22

c

c

vx

2323

18

0724

0But

2

2

2

x

x

x

v

724 22 xv

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

cxv

cxv

22

22

4

221

72

346 i.e.

6,3when 22

c

c

vx

2323

18

0724

0But

2

2

2

x

x

x

v

724 22 xv

212

234,23when

xx

(ii) A particle moves so that its acceleration is given byInitially the particle is 3cm to the right of O and traveling with a velocity of 6cm/s.Find the interval in which the particle moves and determine the greatest acceleration.

xx 4

xvdxd

421 2

cxv

cxv

22

22

4

221

72

346 i.e.

6,3when 22

c

c

vx

2323

18

0724

0But

2

2

2

x

x

x

v

724 22 xv

212

234,23when

xx

2cm/s 212 ison acceleratigreatest

NOTE:

NOTE: At amplitude;v = 0

a is maximum

NOTE: At amplitude;v = 0

a is maximum

At centre;v is maximum

a = 0

Exercise 3D; 1, 6, 7, 10, 12, 14ab, 15ab, 18, 19, 22, 24, 25(start with trig, prove SHM or are told)

Exercise 3F; 1, 4, 5b, 6b, 8, 9a, 10a, 13, 14 a, b(ii,iv), 16, 18, 19(start with )

NOTE: At amplitude;v = 0

a is maximum

At centre;v is maximum

a = 0

xnx 2