āϏ⧂āĻ¤ā§āϰ :

āϏāĻŽā§€āĻ•āĻ°â€ŒāĻŖ

āĻĒā§āϰāϤ⧀āĻ• āĻĒāϰāĻŋāϚāĻŋāϤāĻŋ āĻ“ āĻāĻ•āĻ•

ā§§.āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ• āĻ“ āĻĒāĻ°ā§āϝāĻžā§ŸāĻ•āĻžāϞ⧇āϰ āĻŽāĻ§ā§āϝ⧇ āϏāĻ°ā§āĻŽā§āĻĒāĻ• :n = 1/T

⧍.āϤāϰāĻ™ā§āĻ— āĻŦ⧇āĻ— :v =nÎģ

ā§Š.āϤāϰāĻ™ā§āĻ— āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ :Îģ = vT

ā§Ē.āĻ•ā§ŒāĻŖāĻŋāĻ• āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ•:Ή = 2Ī€/T = 2Ī€n

ā§Ģ.āĻ•āĻŽā§āĻĒāύāĻļā§€āϞ āĻ•āĻŖāĻžāϰ āĻŦāĻžāϰ āĻ•āĻŽā§āĻĒāύ⧇ āϤāϰāĻ™ā§āĻ— āĻ•āĻ°ā§āϤ⧃āĻ• āĻ…āϤāĻŋāĻ•ā§āϰāĻžāĻ¨ā§āϤ āĻĻā§‚āϰāĻ¤ā§āĻŦ:S = NÎģ

ā§Ŧ.āĻ…āĻ—ā§āϰāĻ—āĻžāĻŽā§€ āϤāϰāĻ™ā§āϗ⧇ āϏāĻŽā§€āĻ•āϰāĻŖ :

y=asin2Ī€Îģ(vt−x)
y=asin(Ήt−2Ī€Îģx)
y=asin2Ī€(tT−nÎģ)

ā§­.āĻ¸ā§āĻĨāĻŋāϰ āϤāϰāĻ™ā§āϗ⧇āϰ āϏāĻŽā§€āĻ•āϰāĻŖ :

y=2acos2Ī€xÎģsin2Ī€vtÎģ=Asin2Ī€vtÎģ

ā§Ž.āϤāϰāĻ™ā§āĻ— āĻĒā§āϰāĻŦāĻžāĻšā§‡āϰ āϜāĻ¨ā§āϝ āĻāĻ•āĻ• āĻ†ā§ŸāϤāύ⧇āϰ āĻŽā§‹āϟ āĻļāĻ•ā§āϤāĻŋ :

E = 2Ī€2΁n2a2

⧝.āĻĻ⧁āϟāĻŋ āĻŽāĻžāĻ§ā§āϝāĻŽā§‡ : VAVB=ÎģAÎģB

 

āĻāĻ•āχ āĻŽāĻžāĻ§ā§āϝāĻŽā§‡ : V1V2=Îģ1Îģ2=n2n1

 

ā§§ā§Ļ.āĻĻāĻļāĻž āĻĒāĻžāĻ°ā§āĻĨāĻ•ā§āϝ, Î´=2Ī€Îģ× x āĻĒāĻĨ āĻĒāĻžāĻ°ā§āĻĨāĻ•ā§āϝ 

 

T = āĻĻā§‹āϞāύ āĻ•āĻžāϞ (s)

n = āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ• (Hz)

Îģ = āϤāϰāĻ™ā§āĻ— āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ (m)

Ή = āĻ•ā§ŒāĻŖāĻŋāĻ• āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ• (rads-1)

N = āĻĒā§‚āĻ°ā§āĻŖ āĻ•āĻŽā§āĻĒāύ⧇āϰ āϏāĻ‚āĻ–ā§āϝāĻž

S = āĻĻā§‚āϰāĻ¤ā§āĻŦ (m)

y = āϏāϰāĻŖ(m)

a = āĻŦāĻŋāĻ¸ā§āϤāĻžāϰ (m)

x = āϏāϰāĻŖ(m)

v = āϤāϰāĻ™ā§āĻ— āĻŦ⧇āĻ— (ms-1)

t = āϏāĻŽā§Ÿ (s)

Ή = āϏāĻŽāĻ•ā§ŒāĻŖāĻŋāĻ• āĻŦ⧇āĻ—(ms-1)

΁ = āĻŽāĻžāĻ§ā§āϝāĻŽā§‡āϰ āĻĻā§‚āϰāĻ¤ā§āĻŦ (Kgm3)

E = āĻŽā§‹āϟ āĻļāĻ•ā§āϤāĻŋ (J)

 

 

 

āĻ—āĻžāĻŖāĻŋāϤāĻŋāĻ• āϏāĻŽāĻ¸ā§āϝāĻž āĻ“ āϏāĻŽāĻžāϧāĻžāύāσ

 

ā§§. 224Hz āĻ•āĻŽā§āĻĒāĻžāĻ‚āĻ• āĻŦāĻŋāĻļāĻŋāĻˇā§āϟ āĻāĻ•āϟāĻŋ āϏ⧁āϰ āĻļāϞāĻžāĻ•āĻž āĻšāϤ⧇ āωāϤāĻĒāĻ¨ā§āύ āĻļāĻŦā§āĻĻ āĻŦāĻžāϤāĻžāϏ⧇ 3sec āĻ 1008m āĻĻ⧁āϰāĻ¤ā§āĻŦ āĻ…āϤāĻŋāĻ•ā§āϰāĻŽ āĻ•āϰ⧇ āĨ¤ āϤāϰāĻ™ā§āĻ— āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ āĻŦ⧇āϰ āĻ•āϰ āĨ¤

 

āϏāĻŽāĻžāϧāĻžāύ :

 

S = vt = nÎģt                              S = 1008m

⟹ Îģ = S/nt                               t = 3s

            = 1008/(224×3)                        n = 224Hz

∴ S = 1.5m                   [ans]

 

⧍. āĻŦāĻžāϤāĻžāϏ⧇ āĻļāĻŦā§āĻĻ⧇āϰ āĻŦ⧇āĻ— 332ms-1 āĨ¤ 664Hz āĻ•āĻŽā§āĻĒāĻžāĻ‚āϕ⧇āϰ āϏ⧁āϰ āĻļāϞāĻžāĻ•āĻžāϕ⧇ āĻŦāĻžāϜāĻžāϞ⧇ 100 āĻŦāĻžāϰ āĻ•āĻŽā§āĻĒāύ āĻ•āĻžāϞ⧇ āĻ•āϤ āĻĻā§‚āϰāĻ¤ā§āĻŦ āĻ…āϤāĻŋāĻ•ā§āϰāĻŽ āĻ•āϰāĻŦ⧇ āĨ¤

 

āϏāĻŽāĻžāϧāĻžāύ :

 

S = NÎģ                                      n = 664Hz

  = N(v/n)                                 v = 332 ms-1  

  = 100×(332/664)                    N = 100

∴ S = 50m                    [ans]

 

 

ā§Š. āϤāϰāĻ™ā§āĻ—āĻ¸ā§āĻĨāĻŋāϤ 0.297m  āĻŦā§āϝāĻŦāϧāĻžāύ⧇  āĻ…āĻŦāĻ¸ā§āĻĨāĻŋāϤ āĻĻ⧁āϟāĻŋ āĻ•āύāĻžāϰ āĻŽāĻ§ā§āϝ⧇ āĻĻāĻļāĻž āĻĒāĻžāĻ°ā§āĻĨāĻ•ā§āϝ 1.57 āϰ⧇āĻĄāĻŋ⧟āĻžāύ āĨ¤ āϤāϰāĻ™ā§āĻ— āĻ‰ā§ŽāϏ⧇āϰ āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ• 280Hz āĻšāϞ⧇ āϤāϰāĻ™ā§āĻ— āĻŦ⧇āĻ— āĻ•āϤ?

 

āϏāĻŽāĻžāϧāĻžāύ :

 

θ = (2Ī€x)/Îģ                                           n = 280Hz

⟹ Îģ = (2Ī€x)/θ                                     x = .297m

⟹ Îģ = (2Ī€Ã—.297)/1.57 = 1.188m         θ = 1.57rad

∴ v = nÎą = 280×1.188

∴ v = 332.6ms-1                       [ans]

 

ā§Ē. āĻāĻ•āϟāĻŋ āϏ⧁āϰāĻļāϞāĻžāĻ•āĻžāϰ āĻŦāĻžāĻšā§āĻĻā§āĻŦ⧟ āϝ⧇ āϏāĻŽā§Ÿā§‡ 200 āĻŦāĻžāϰ āĻ•āĻŽā§āĻĒāύ āϏāĻŽā§āĻĒāĻ¨ā§āύ āĻ•āϰ⧇,āϏ⧇ āϏāĻŽā§Ÿā§‡ āĻāϟāĻŋ āĻĻā§āĻŦāĻžāϰāĻž āϏ⧃āĻˇā§āϟāĻŋ āĻļāĻŦā§āĻĻ āϤāϰāĻ™ā§āĻ— āĻŦāĻžāϤāĻžāϏ⧇ 140m  āĻĻā§‚āϰāĻ¤ā§āĻŦ āĻ…āϤāĻŋāĻ•ā§āϰāĻŽ āĻ•āϰ⧇ āĨ¤ āĻļāϞāĻžāĻ•āĻžāϟāĻŋāϰ āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ• 490Hz āĻšāϞ⧇ āĻŦāĻžāϤāĻžāϏ⧇āϰ āĻŽāĻ§ā§āϝ⧇ āĻļāĻŦā§āĻĻ⧇āϰ āĻŦ⧇āĻ— āύāĻŋāĻ°ā§āύ⧟ āĻ•āϰ āĨ¤

 

āϏāĻŽāĻžāϧāĻžāύ :

 

v = nÎģ = n(S/N)                                                n = 490Hz

⟹ v = 490×(140/200)                         S = 140m

∴ v = 343ms-1              [ans]                N = 200

 

ā§Ģ. Y = 10 sin2(t/0.02 – x/15) āϏāĻŽā§€āĻ•āϰāĻŖāϟāĻŋ āĻāĻ•āϟāĻŋ āĻ…āĻ—ā§āϰāĻ—āĻžāĻŽā§€ āϤāϰāĻ™ā§āĻ— āĻĒā§āϰāĻ•āĻžāĻļ āĻ•āϰ⧇ āĨ¤ āĻĻ⧈āĻ°ā§āĻ˜ā§āϝāϰ āĻāĻ•āĻ• cm āĻ“ āϏāĻŽā§Ÿā§‡āϰ sec āĨ¤ āϤāϰāĻ™ā§āϗ⧇āϰ āĻŦāĻŋāĻ¸ā§āϤāĻžāϰ, āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ•, āϤāϰāĻ™ā§āĻ— āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ āĻāĻŦāĻ‚ āϤāϰāĻ™ā§āϗ⧇āϰ āĻŦ⧇āĻ— āύāĻŋāĻ°ā§āϪ⧟ āĻ•āϰ āĨ¤  

 

āϏāĻŽāĻžāϧāĻžāύ :

 

y = 10 sin2(t/0.02 – x/15)

⟹ y = 10 sin2(5ot - x/15)

            = 10 sin(2Ī€/15)(750t-x)...(i)

we know,

            y = a sin(2Ī€/Îą)(vt-x)...(ii)

from equation (i) & (ii) ⟹

āĻŦāĻŋāĻ¸ā§āϤāĻžāϰ, a = 10cm                                                [ans]

āϤāϰāĻ™ā§āĻ— āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ, Îģ = 15cm                             [ans]

āĻŦ⧇āĻ—, v = 750mcs-1                                [ans]

āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ•, n = v/Îģ = 750/15 = 50Hz                      [ans]