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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)
xx
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)
xxkxx
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)
xxkxx
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)
xxkxx
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,undergoesparticle 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,undergoesparticle 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,undergoesparticle 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
xanvanxnv
when x = a , v = 0 (a = amplitude)
22
2220 i.e.anc
can
22
2222
22222
xanv
xanvanxnv
NOTE: 02 v
when x = a , v = 0 (a = amplitude)
22
2220 i.e.anc
can
22
2222
22222
xanv
xanvanxnv
NOTE: 02 v
022 xa
when x = a , v = 0 (a = amplitude)
22
2220 i.e.anc
can
22
2222
22222
xanv
xanvanxnv
NOTE: 02 v
022 xaaxa
when x = a , v = 0 (a = amplitude)
22
2220 i.e.anc
can
22
2222
22222
xanv
xanvanxnv
NOTE: 02 v
022 xaaxa
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 ,0cos and ve- choose ,
;0 when If
xax
t
22 xandtdx
1-
1-
sin and ve choose ,0cos and ve- choose ,
;0 when If
xax
t
22
1xandx
dt
22 xandtdx
1-
1-
sin and ve choose ,0cos and ve- choose ,
;0 when If
xax
t
22
1xandx
dt
dxxan
tx
a
22
11
22 xandtdx
1-
1-
sin and ve choose ,0cos and ve- choose ,
;0 when If
xax
t
22
1xandx
dt
dxxan
tx
a
22
11
x
aax
n
1cos1
22 xandtdx
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1-
sin and ve choose ,0cos and ve- choose ,
;0 when If
xax
t
22
1xandx
dt
dxxan
tx
a
22
11
x
aax
n
1cos1
ax
n
ax
n1
11
cos1
1coscos1
22 xandtdx
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1-
sin and ve choose ,0cos and ve- choose ,
;0 when If
xax
t
22
1xandx
dt
dxxan
tx
a
22
11
x
aax
n
1cos1
ax
n
ax
n1
11
cos1
1coscos1
ntax
ntax
axnt
cos
cos
cos 1
In general;
In general;
xnx 2
obeysSHMundergoing particleA
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
ntax cos
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
ntax cosamplitude where
sinOR
a
ntax
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
ntax cosamplitude where
sinOR
a
ntax
the particle has;
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
ntax cosamplitude where
sinOR
a
ntax
the particle has;
Tf
nT
1 :frequency
2 :period
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
ntax cosamplitude where
sinOR
a
ntax
the particle has;
Tf
nT
1 :frequency
2 :period
(time for one oscillation)
In general;
xnx 2
obeysSHMundergoing particleA
2222 xanv particletheofpath find tous allows
ntax cosamplitude where
sinOR
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.
txtx
tx
3sin363cos12
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.
txtx
tx
3sin363cos12
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.
txtx
tx
3sin363cos12
3sin4
x9SHMin movesparticle
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.
txtx
tx
3sin363cos12
3sin4
x9SHMin movesparticle
32
T
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.
txtx
tx
3sin363cos12
3sin4
x9SHMin movesparticle
32
T
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.
txtx
tx
3sin363cos12
3sin4
x9SHMin movesparticle
32
T
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.
txtx
tx
3sin363cos12
3sin4
x9SHMin movesparticle
32
T
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 44interval thealongmovesparticle 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 4
21 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 4
21 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 4
21 2
cxv
cxv
22
22
4
221
72
346 i.e.
6,3when 22
cc
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 4
21 2
cxv
cxv
22
22
4
221
72
346 i.e.
6,3when 22
cc
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 4
21 2
cxv
cxv
22
22
4
221
72
346 i.e.
6,3when 22
cc
vx
232318
07240But
2
2
2
xxx
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 4
21 2
cxv
cxv
22
22
4
221
72
346 i.e.
6,3when 22
cc
vx
232318
07240But
2
2
2
xxx
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 4
21 2
cxv
cxv
22
22
4
221
72
346 i.e.
6,3when 22
cc
vx
232318
07240But
2
2
2
xxx
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