2. Simple Harmonic Motion
A 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)
3. Simple Harmonic Motion
A 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)
x
x
4. Simple Harmonic Motion
A 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)
x
x
kx
x
5. Simple Harmonic Motion
A 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)
x
x
kx
x
n 2 x
x
6. Simple Harmonic Motion
A 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)
x
x
kx
x
n 2 x (constant needs to be negative)
x
7. Simple Harmonic Motion
A 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)
x
x
kx
x
n 2 x (constant needs to be negative)
x
If a particle undergoes SHM, then it obeys;
n 2 x
x
8. Simple Harmonic Motion
A 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)
x
x
kx
x
n 2 x (constant needs to be negative)
x
If a particle undergoes SHM, then it obeys;
n 2 x
x
d 1 2
v n x
2
dx 2
9. Simple Harmonic Motion
A 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)
x
x
kx
x
n 2 x (constant needs to be negative)
x
If a particle undergoes SHM, then it obeys;
n 2 x
x
d 1 2
v n x
2
dx 2
1 2 1 2 2
v n x c
2 2
v 2 n 2 x 2 c
11. when x = a , v = 0 (a = amplitude)
i.e. 0 2 n 2 a 2 c
c n2a 2
12. when x = a , v = 0 (a = amplitude)
i.e. 0 2 n 2 a 2 c
c n2a 2
v 2 n 2 x 2 n 2 a 2
v 2 n 2 a 2 x 2
v n a 2 x 2
13. when x = a , v = 0 (a = amplitude)
i.e. 0 2 n 2 a 2 c
c n2a 2
v 2 n 2 x 2 n 2 a 2
v 2 n 2 a 2 x 2
v n a 2 x 2
NOTE: v2 0
14. when x = a , v = 0 (a = amplitude)
i.e. 0 2 n 2 a 2 c
c n2a 2
v 2 n 2 x 2 n 2 a 2
v 2 n 2 a 2 x 2
v n a 2 x 2
NOTE: v2 0
a2 x2 0
15. when x = a , v = 0 (a = amplitude)
i.e. 0 2 n 2 a 2 c
c n2a 2
v 2 n 2 x 2 n 2 a 2
v 2 n 2 a 2 x 2
v n a 2 x 2
NOTE: v2 0
a2 x2 0
a x a
16. when x = a , v = 0 (a = amplitude)
i.e. 0 2 n 2 a 2 c
c n2a 2
v 2 n 2 x 2 n 2 a 2
v 2 n 2 a 2 x 2
v n a 2 x 2
NOTE: v2 0
a2 x2 0
a x a
Particle travels back and forward between x = -a and x = a
18. dx
n a 2 x 2
dt If when t 0;
x a, choose - ve and cos -1
x 0, choose ve and sin -1
19. dx
n a 2 x 2
dt If when t 0;
dt 1 x a, choose - ve and cos -1
dx n a 2 x 2
x 0, choose ve and sin -1
20. dx
n a 2 x 2
dt If when t 0;
dt 1 x a, choose - ve and cos -1
dx n a 2 x 2
x 0, choose ve and sin -1
1
x
1
t 2 dx
na a x 2
21. dx
n a 2 x 2
dt If when t 0;
dt 1 x a, choose - ve and cos -1
dx n a 2 x 2
x 0, choose ve and sin -1
1
x
1
t 2 dx
na a x 2
x
1 x
cos 1
n aa
22. dx
n a 2 x 2
dt If when t 0;
dt 1 x a, choose - ve and cos -1
dx n a 2 x 2
x 0, choose ve and sin -1
1
x
1
t 2 dx
na a x 2
x
1 x
cos 1
n aa
1 1 x
cos cos 1 1
n a
1 1 x
cos
n a
23. dx
n a 2 x 2
dt If when t 0;
dt 1 x a, choose - ve and cos -1
dx n a 2 x 2
x 0, choose ve and sin -1
1
x
1
t 2 dx
na a x 2
x
1 x
cos 1
n aa
1 1 x
cos cos 1 1
n a
1 1 x
cos
n a
1 x
nt cos
a
x
cos nt
a
x a cos nt
25. In general;
A particle undergoing SHM obeys
n 2 x
x
26. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2
27. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
28. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
x a cos nt
29. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
x a cos nt OR x a sin nt
where a amplitude
30. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
x a cos nt OR x a sin nt
where a amplitude
the particle has;
31. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
x a cos nt OR x a sin nt
where a amplitude
the particle has;
2
period : T
n
1
frequency : f
T
32. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
x a cos nt OR x a sin nt
where a amplitude
the particle has;
2
period : T (time for one oscillation)
n
1
frequency : f
T
33. In general;
A particle undergoing SHM obeys
n 2 x
x
v 2 n 2 a 2 x 2 allows us to find path of the particle
x a cos nt OR x a sin nt
where a amplitude
the particle has;
2
period : T (time for one oscillation)
n
1
frequency : f (number of oscillations
T per time period)
34. e.g. (i) A particle, P, moves on the x axis according to the law x = 4sin3t.
35. 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.
36. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
37. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
9 x
38. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
9 x
particle moves in SHM
39. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
9 x
particle moves in SHM
2
T
3
40. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
9 x
particle moves in SHM
2
T
3
2
period of motion is seconds
3
41. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
9 x
particle moves in SHM
2
T
3
2
period of motion is seconds
3
b) Find the interval in which the particle moves and determine the
greatest speed.
42. 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.
x 4 sin 3t
x 12 cos 3t
36 sin 3t
x
9 x
particle moves in SHM
2
T
3
2
period of motion is seconds
3
b) Find the interval in which the particle moves and determine the
greatest speed.
particle moves along the interval 4 x 4
and the greatest speed is 12 units/s
43. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
44. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2
v 4 x
dx 2
45. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2
v 4 x
dx 2
1 2
v 2 x 2 c
2
v 2 4 x 2 c
46. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2
v 4 x
dx 2
1 2
v 2 x 2 c
2
v 2 4 x 2 c
when x 3, v 6
i.e. 6 2 43 c
2
c 72
47. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2
v 4 x
dx 2
1 2
v 2 x 2 c
2
v 2 4 x 2 c
when x 3, v 6
i.e. 6 2 43 c
2
c 72
v 2 4 x 2 72
48. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2 But v 2 0
v 4 x
dx 2 4 x 2 72 0
1 2 x 2 18
v 2 x 2 c
2
3 2 x 3 2
v 2 4 x 2 c
when x 3, v 6
i.e. 6 2 43 c
2
c 72
v 2 4 x 2 72
49. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2 But v 2 0
v 4 x
dx 2 4 x 2 72 0
1 2 x 2 18
v 2 x 2 c
2
3 2 x 3 2
v 2 4 x 2 c
when x 3, v 6 when x 3 2, 43 2
x
i.e. 6 2 43 c
2 12 2
c 72
v 2 4 x 2 72
50. (ii) A particle moves so that its acceleration is given by 4 x
x
Initially 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.
d 1 2 But v 2 0
v 4 x
dx 2 4 x 2 72 0
1 2 x 2 18
v 2 x 2 c
2
3 2 x 3 2
v 2 4 x 2 c
when x 3, v 6 when x 3 2, 43 2
x
i.e. 6 2 43 c
2 12 2
c 72 greatest acceleration is 12 2 cm/s 2
v 2 4 x 2 72
53. NOTE: At amplitude;
v=0
a is maximum
At centre;
v is maximum
a=0
54. 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 n 2 x)
x