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11
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2. 11
F
F F F F ˆˉ
F
F F
F
- F F F F F F
- S.H.M. ˈ
y = A sin ߱t
y= y
A=
߱ =
t= F
FF F F y = A sin ߱t F S.H.M
y F
r
θ
y = r sin θ
r=A θ = ߱t
F y = A sin ߱t
– F ˆ F www.schoolDD.com 1

3. F F
t=0
A
t = T/4
t = T/2
t = 3T/4
A
t=T
F F F
y
y = A sin ߱t
A
0 t
T/4 T/2 3T/4 T
-A
11.1
1. ˈ F F
F
2. F ˂ ˈ F F
F ˈ 2
1.
ˈ F
F
2.
ˈ F
– F ˆ F www.schoolDD.com 2

4. 11.2
ˈ F F ˈ
λ
A A’
O
P
B F B’ F
λ
- F F F O, P
- F
F F A B
- F F F A A
- F F F F B B
- ૃ F F F F
F AA , BB F F F F 2π
F F ˈ (m)
ߣ ߣ ߣ ߣ
ߣ
F F λ
F ( ) F 1
– F ˆ F www.schoolDD.com 3

5. 0
t=0
λ/4
t = T/4
λ/2
t = T/2
3λ/4
t = 3T/4
λ
t=T
11.3
F F F ˈ /
(1) ୱ
= v=
୲
t s F λ F
F T
ୱ
‫׵‬
ߣ
= fλ
(2)
୲ ୘
v= =
v=fૃ
v =
f = F F F
F ˈ / F (Hz)
T= F F F F ˈ ˈ
F
λ= F ˈ
- F F
– F ˆ F www.schoolDD.com 4

6. ˈ F F F F F F ˈ
A 0°
గ
B B 90° ଶ
A C 180° ߨ
C E ଷగ
D 270° ଶ
D
E 360° 2ߨ
- F F λ F 360˚ 2ߨ
2 F
1. F 2π, 4π, 6π,
2. F λ , 2λ , 3λ ,
3. F T, 2T, 3T,
B F J A,E,I M
P R B,F,J
A C G K
Q
E
S
I M C,G,K
D H L D,H,L
P,R
Q,S
F 2 F F F
F
1. F π, 3π, 5π,
2. F λ/2, 3λ/2, 5λ/2,
3. F T/2, 3T/2, 5T/2,
C,G,K F A
D,H,L F B
A,E,I,M F C
Q,S F P
– F ˆ F www.schoolDD.com 5

7. F 1
F F
(cm)
+10
0
a b (cm)
5 10 15 20
-10
. λ
. A
. a b
. λ=?
λ = 10 cm = 0.10 m Ans
.
= 10 cm = 0.10 m Ans
.
a = π Ans
b = 2π Ans
F 2
F F F F F 30
10 F F ˈ 4 cm
4 cm 30 / 10
. ଷ଴
f= =
ଵ଴
‫ ׵‬f = 3.0 Hz Ans
v = f λ = 3.0 x 0.04
‫ ׵‬v = 0.12 m/s Ans
– F ˆ F www.schoolDD.com 6

8. ଵ ଵ
T= =
୤ ଷ
‫ ׵‬T = 0.33 s Ans
F 3
A ˈ B F 2
cm
A 3
1 2 4 5 6
cm
B 1 2 3 4 5 6
F A B
s = 6-3 = 3 cm
F t=2 s
ୱ ଷ.଴
‫׵‬ = 1.5 m/s Ans
୲ ଶ
v= =
f v = fλ
λ = 4 cm
ଵ.ହ
‫׵‬f=
୴
= 0.375 Hz Ans
ସ
=
஛
F 4
F F 20 cm/s 5 / F
F F 1.0 m F F F
ˁ
1.0
F F v = 20 cm/s , f = 5 Hz
λ v = fλ
– F ˆ F www.schoolDD.com 7

9. ଶ଴
‫׵‬λ=
୴
= 4.0 cm = 0.04 m Ans
ହ
=
୤
F F
ଷ஛
s = 1.0 + ସ
= 1.03 m
ୱ
୲
v=
ୱ ଵ.଴ଷ
‫׵‬ = 5.15 s Ans
୴ ଴.ଶ଴
t= =
F 5
F 100 / F 0.5 F ߨ/4
0.50
0 ߨ/4
F ߨ/4 F F
0.5
F 2ߨ 0.5x2ߨ/( ߨ/4) = 4.0
F F
‫ ׵‬λ = 4.0 m Ans
v v = f λ = 100x4.0
‫ ׵‬v = 400 m/s Ans
11.4
ˈ F F
F F F
– F ˆ F www.schoolDD.com 8

10. - F ˈ F ( - F )
- ˈ F F
11.5 F
ˈ F F F
1 F F
F ˈ ˈ F F
F F
- F F
11.5 F
F F F
F
F
F F F
F F
11.6 F
F ˈ F
F F F F F F F
F
– F ˆ F www.schoolDD.com 9

11. F
F
F 6
2 F F 1 cm/s F F
F
cm
+1
cm
1 2 3 4 5 6
-1
F 2 cm
cm
+2
+1
cm
1 2 3 4 5 6
-1
-2
F s = 2 cm
t F s= vt
ୱ ଶ.଴
‫׵‬ = 2.0 s Ans
୴ ଵ.଴
t= =
= 1.0 + 1.0 = 2.0 m Ans
– F ˆ F www.schoolDD.com 10

12. 11.7
4 F
- F ˈ F F
ˈ F
11.8 F
1. F F
F F F F
F F F
F F
ˈ F F F
F
F
F 7
F F 1 cm/s F F
ˈ F F F 2 cm
– F ˆ F www.schoolDD.com 11

13. F F ˈ
F
F
F ˈ F
F F 2 cm
t F s= vt
ୱ ଶ.଴
‫׵‬ = 2.0 s Ans
୴ ଵ.଴
t= =
2. F
F F F F F F F
F F F F F
F
F
F F
F
θ1 θ2
θ1 θ2 F
ી1 = F ી2 F
θ1 , θ2 = F F F
F = F F
- F F F
F 8
F F F
50°
120°
– F ˆ F www.schoolDD.com 12

14. 11.9
F F
F
F
F F F
C λ1 B
θ2
1
θ1 θ1
2
A θ2 λ2
D
୆େ ஛భ
sin θଵ = = ------
୅୆ AB
୅ୈ ஛మ
sin θଶ = = ------
୅୆ AB
ୱ୧୬ ஘భ ஛భ ୚భ /୤భ ୴భ
‫ ܖܑܛ‬ી૚ ૃ૚ ‫ܞ‬૚ / ,
ୱ୧୬ ஘మ
= ஛మ
= ୚మ /୤మ
=
୴మ
‫ ܖܑܛ‬ી૛ ૃ૛ ‫ܞ‬૛
= =
F f1 = f2
θ1 =
= F
θ2 =
- F F F F ˈ
F
v >v
ୱ୧୬ ஘భ ୴
ୱ୧୬ ஘మ ୴
=
‫ ׵‬sin θଵ > sin θଶ
F θ1 > θ2 ( F sin θ θ )
θ1
F F F F F
θ2
- F F
– F ˆ F www.schoolDD.com 13

15. F 9
F 1 m/s F F F
ˈ 0.50 m 1.0 m
θ1
v1 = 1 m/s θ2 v2 = ?
λ1 = 0.5 m λ2 = 1.0 m
୴భ ஛భ
୴మ ஛మ
=
ଵ ଴.ହ
F F ୴మ
=
ଵ.଴
‫ ׵‬v2 = 2 m/s Ans
f v=fλ
୴భ ୴మ ଶ.଴
f=
஛భ
=
஛మ
=
ଵ.଴
= 2.0 Hz Ans
F 10
F 2.0 cm 3 Hz 1 F 2 F
16˚ 2
2.0 cm
1
53° 2
16°
θ1 = 53°
θ2 = 53° - 16° = 37°
ୱ୧୬ ஘భ ஛భ
ୱ୧୬ ஘మ ஛మ
=
ୱ୧୬ ହଷ ஛భ
ୱ୧୬ ଷ଻ ஛మ
=
ସ/ହ ଶ.଴
F F ଷ/ହ
=
஛మ
– F ˆ F www.schoolDD.com 14

16. ‫ ׵‬λଶ = 1.5 m Ans
v2 v = fλ
v2 = 3x1.5 = 4.5 m/s Ans
11.10
ˈ F2 F F ( F
)
F F
N1 A0 N1
A1 A1
N2 N2
A2 A2
N3 N3
F
A3 A3
s1 s2
s1 s2 ˈ F F
ˈ F ( )
ˈ F F F ( )
ˈ F F ( )
(Antinode) F
(Node) F F ˈ F
F A0 , A1 , A2 , , An ˈ F F F F
F N1 , N2 , N3 , , Nn ˈ F F F F
– F ˆ F www.schoolDD.com 15

17. F F F s1 , s2 F F
A0
A1 N1 N1 A1
N2 N2
Q
P
s1 + + s2
d
n = 0, 1, 2, 3,
| s 1P – s 2P | = n ૃ P F F n ( An )
n=0 ˈ F ( A0 )
n = 1, 2, 3,
૚
| s 1Q – s 2Q | = ( n - ) ૃ Q F F n ( Nn )
૛ n= 1 F
F F
N1 A0 N1
A1 A1
N2 N2
A2 P1 A2
Q1
P2 N3
N3
F
Q3
P3
A3 Q2 A3
+ +
s1 s2
– F ˆ F www.schoolDD.com 16

18. F P1 , | s1P1 s2P1 | = | 5 λ - 5 λ | = 0 λ ‫׵‬n = 0
F P1 ˈ F F (A0)
F P2 , | s1P2 s2P2 | = | 5 λ - 4 λ | = 1 λ ‫׵‬n = 1
F P2 ˈ F F 1 (A1)
F P3 , | s1P3 s2P3 | = | 1.5 λ 3.5 λ | = 2 λ ‫ ׵‬n = 2
F P3 ˈ F F 2 (A2)
F Q1 , | s1Q1 s2Q1 | = | 4.5 λ - 5 λ | = 0.5 λ ‫ ׵‬n-1/2 = 0.5 n=1
F Q1 ˈ F F F 1 (N1)
F Q2 , | s1Q2 s2Q2 | = | 1.5 λ - 4 λ | = 2.5 λ ‫ ׵‬n-1/2 = 2.5 n=3
F Q2 ˈ F F F 3 (N3)
F Q3 , | s1Q3 s2Q3 | = | 4 λ 2.5 λ | = 1.5 λ ‫ ׵‬n-1/2 = 1.5 n=2
F Q3 ˈ F F F 2 (N2)
F X , | s1X s2X | = | 5 λ 2.5 λ | = ‫׵‬ n= .
F X ˈ F ................... F ............ .......( ..)
F F F s1 s2 ,
F
d=nૃ d = ( n- ) ૃ
૚
૛
N1 A0 N1
A1 A1
N2 N2
A2 A2
N3 N3
An An
+ +
s1 s2
d F
– F ˆ F www.schoolDD.com 17

19. 1. F ˈ An F n d=nλ
2. F F n (n = 0, 1, 2, ) F F
3. F F n F ( n = 0.5, 1.5, 2.5, ) F F ˈ Nn F F n F
F d = (n - ଵ) λ
ଶ
F
- d=nλ d = (n - ଵ) λ
ଶ
F P Q F F s2 F
s1P s2P = d s1Q s2Q = d ˈ F
- F n ˈ F F F F
s1 s2 ˈ A0 FF d λ F F
F 11
s1 s2 ˈ F F F F 10 cm F 2 cm
F F F F F ˈ 8 cm 13 cm
F F
P, Q
8 cm 13 cm
λ = 2 cm
+ +
s1 d = 10 cm s2
F F F F
F n F | s1P s2P | = n λ ( F F n )
| 8 13 | = n (2)
n = 2.5
F n F F F F F F F
F n F F | s1Q s2Q | = (n - ଵ) λ
ଶ
| 8 13 | = (n - ଵ) (2)
ଶ
n = 3.0
‫׵‬ F F F F 3 (N3 ) Ans
– F ˆ F www.schoolDD.com 18

20. F
F ˈ F An
F n F d = nλ ( F F n )
10 = n (2)
n = 5 (n ‫׵‬ F )
F ˈ 5 (A5 )
A0 N1 A1
N2
A2
N3
A3
N4 11
A4
N5
10 Ans
A5
F F ”
+ + F
s1 d s2
F 12
F F s1 , s2 F F 7 cm F F 2 cm
F F s1 s2
A0
λ = 2 cm
+ +
s1 d =7 cm s2
F F F s1 s2 ˈ
F n F d = nλ ( F F n )
7 = n (2)
n = 3.5 ( n F ‫׵‬ )
F F F F ˈ
– F ˆ F www.schoolDD.com 19

21. F n d = (n - ଵ) λ
ଶ
7 = (n - ଵ) (2)
ଶ
n=4
F F F s1 s2 ˈ 4 (N4 )
F F s1 s2 F F
F 13
F F s1 , s2 F F 6 cm 1 cm M F F s1 s2
ˈ 12 cm 8 cm F F s2 M
A0
M
12 cm 8 cm
λ = 1 cm
+ +
s1 d =6 cm s2
F F F M s2 ˈ
F M ˈ
F n F | s1M s2M | = n λ ( F F n )
| 12 8 | = n (1)
n = 4 (n ‫׵‬ F )
F M F F 4 (A4 )
F s2 ˈ
F n F d = nλ ( F F n )
6 = n (1)
n = 6 (n ‫׵‬ F )
– F ˆ F www.schoolDD.com 20

22. F s2 F F 6 (A6 )
F F s2M F
11.11
ˈ 2 F F
F F
. F
∫
λ/2
λ/2 1 Loop
.
∫
– F ˆ F www.schoolDD.com 21

23. - F F ˈ
- F ˈ
F 14
4.5 F F ˈ
F 1 m/s
4.5
F 2.25 λ = 4.5 m
‫ ׵‬λ = 2.0 m Ans
f v = fλ
‫ ׵‬f = ஛ = ଵ.଴ = 0.5 Hz Ans
୴
ଶ.଴
F 15
F F F F 8 cm F F
F F F F 8 F
8.0 cm
ˈ F F
8 F F loop F 7 loop ½ loop F 8
loop
8 loop = 8.0 cm
1 loop = 1.0 cm
F 1 loop = λ/2
‫ ׵‬λ = 2 x 1.0 = 2.0 m Ans
– F ˆ F www.schoolDD.com 22

24. 11.12
ˈ F F F F
- F
- F
- F F ʽ F F ʽ (d) F F
(λ )
λ λ
d >λ d <λ
- F F ʽ F F ˈ ˈ
F F F F
F F F F F FF ˈ F F F
F F
F F
F F
F
F
– F ˆ F www.schoolDD.com 23

25. - F F
ˈ F
- F F ʽ d >λ ˈ
- F F ʽ d< λ ˈ
N N
N A A
N
A0 A A0 A
11.13 F (Resonance)
ˈ F F F ˈ F
F F F
F F F F F
F F F F
F
F F F F
ଵ ௚ ଵ ୩
f= ට f= ට
ଶగ ℓ ଶగ ୫
F F
ℓ
ℓ
T = 2ߨට௚ F f = ୘ = ଶగ ට௚
ଵ ଵ
ℓ
ଵ
f ‫ן‬ g
√ℓ
ଵ ଵ ୏
k
T = 2ߨට୫
୏
F f = ୘ = ଶగ ට୫
ଵ
k f ‫ן‬ K
m √୫
m
– F ˆ F www.schoolDD.com 24

26. F
F F F F
F
஛భ ୴ ୴
ଶ
= L , f1 = ஛ = ଶ୐ = F 1
భ
୴ ୴
λଶ = L , f2 = = = F 2
஛మ ୐
୴ ୴
λଶ
஛ ୐
ଷ஛య ୴ ଷ୴
ଶ
= L , f3 = ஛ = ଶ୐ = F 3
య
L
୬୴
fn =
ଶ୐
= F n
fn = F F F
n = loop
- F F F F F F F ( )
F F F F
– F ˆ F www.schoolDD.com 25