āϏāĻŽā§€āĻ•āϰāĻŖ

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

ā§§.āĻĒā§āϰāĻ¤ā§āϝāĻžā§Ÿāύ⧀ āĻŦāϞ,F = -Kx

 

⧍.āϏāϰāϞ āĻ›āĻ¨ā§āĻĻāĻŋāϤ āĻ—āϤāĻŋāϰ āĻ•ā§āώ⧇āĻ¤ā§āϰ⧇ :

āĻ—āϤāĻŋāϰ āϏāĻŽā§€āĻ•āϰāĻŖ : d2xdt2+Ή2x=0

āϏāϰāϪ⧇āϰ āϏāĻŽā§€āĻ•āϰāĻŖ : x = Asin (Ήt+δ )

 

ā§Š.āĻ•ā§ŒāϞāĻŋāĻ• āĻŦ⧇āĻ—, Ī‰=Km

 

ā§Ē.āĻĻā§‹āϞāύāĻ•āĻžāϞ, T=2΀Ή=2Ī€mK

 

ā§Ģ. āĻ•āĻŽāĻĒāĻžāĻ™ā§āĻ• : n=12Ī€Km

ā§Ŧ. āĻŦ⧇āĻ— : v=ΉA2−x2

ā§­. āĻ¤ā§āĻŦāϰāĻŖ : a = - Ή2A

ā§Ž. āϏāĻ°ā§āĻŦā§‹āĻšā§āϚ āĻŦ⧇āĻ— :vmax = ΉA

⧝. āϏāĻ°ā§āĻŦā§‹āĻšā§āϚ āĻ¤ā§āĻŦāϰāĻŖ :amax = Ή2A

ā§§ā§Ļ. āĻ—āϤāĻŋ āĻļāĻ•ā§āϤāĻŋ :k = ÂŊ K (A2 – x2)

ā§§ā§§.āĻ¸ā§āĻĨāĻŋāϤāĻŋ āĻļāĻ•ā§āϤāĻŋ :U = ÂŊ kx2

⧧⧍. āĻ¸ā§āĻĒā§āϰāĻŋāĻ‚ā§Ÿā§‡āϰ āĻĻā§‹āϞāύāĻžāĻ•āĻžāϞ : T=2Ī€mk

ā§§ā§Š. āĻ¸ā§āĻĒā§āϰāĻŋāĻ‚ā§Ÿā§‡āϰ āĻŦāϞ āĻ§ā§āϰ⧁āĻŦāĻ• :K = mg /I

ā§§ā§Ē. āϏāϰāϞ āĻĻā§‹āϞāϕ⧇ āĻĻā§‹āϞāύāĻ•āĻžāϞ : T=2Ī€Lg

ā§§ā§Ģ. āĻĻā§‹āϞāύāĻ•āĻžāϞ :  T = I /n

 

F = āĻĒā§āϰāĻ¤ā§āϝ⧟āύ⧀ āĻŦāϞ

K = āĻŦāϞ āĻ§ā§āϰ⧁āĻŦāĻ•

x = āϏāϰāĻŖ

A = āĻŦāĻŋāĻ¸ā§āϤāĻžāϰ

t = āϏāĻŽā§Ÿ

δ = āφāĻĻāĻŋ āĻĻāĻļāĻž

Ή = āĻ•ā§ŒāĻŖāĻŋāĻ• āĻ•āĻŽā§āĻĒāĻžāĻ™ā§āĻ•

m = āĻ­āϰ

T = āĻĻā§‹āϞāύāĻ•āĻžāϞ

v = āĻŦ⧇āĻ—

a = āĻ¤ā§āĻŦāϰāĻŖ

g = āĻ…āĻ­āĻŋāĻ•āĻ°ā§āĻˇā§€ā§Ÿ āĻ¤ā§āĻŦāϰāĻŖ

l= āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ

L = āĻĻā§‹āϞāϕ⧇āϰ āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ

n = āĻ•āĻŽāĻĒāĻžāĻ™ā§āĻ•

 

 

 

 

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

 

  • Magic → T1/T2 = √(g2/g1) = (86400-x)/(86400)

→ R/(R+h) = √{(R-h)/R} = √(L1/L2)

  • āϞāĻŋāĻĢāĻŸā§‡āϰ āĻ•ā§āώ⧇āĻ¤ā§āϰ⧇, āύāĻŋāĻšā§‡āϰ āĻĻāĻŋāϕ⧇, T = 2Ī€ √{L/(g-t)}

āωāĻĒāϰ⧇āϰ āĻĻāĻŋāϕ⧇, T = 2Ī€ √{L/(g+t)}

 

ā§§. 30Nm-1 āĻ§ā§āϰ⧁āĻŦāϕ⧇āϰ āĻāĻ•āϟāĻŋ āφāύ⧁āĻ­ā§‚āĻŽāĻŋāĻ• āĻ¸ā§āĻĒā§āϰāĻŋāĻ‚ āĻāϰ āĻāĻ• āĻĒā§āϰāĻžāĻ¨ā§āϤ āĻāĻ•āϟāĻŋ āĻĻā§‡ā§ŸāĻžāϞ⧇āϰ āϏāĻžāĻĨ⧇ āφāϟāĻ•āĻŋā§Ÿā§‡ āĻ…āĻĒāϰ āĻĒā§āϰāĻžāĻ¨ā§āϤ⧇ 0.5kg-wt āĻ“āϜāύ⧇āϰ āĻāĻ•āϟāĻŋ āĻŦā§āϞāĻ• āφāϟāĻ•āĻŋā§Ÿā§‡ āϏāĻžāĻŽā§āϝāĻžāĻŦāĻ¸ā§āĻĨāĻžāύ āĻĨ⧇āϕ⧇ āĻāĻ•āϟāĻŋ āφāύ⧁āĻ­ā§‚āĻŽāĻŋāĻ• āϘāĻ°ā§āώāĻŖāĻšā§€āύ āĻŸā§‡āĻŦāĻŋāϞ āĻŦāϰāĻžāĻŦāϰ 10cm āĻŸā§‡āύ⧇ āĻ›ā§‡ā§œā§‡ āĻĻā§‡ā§ŸāĻž āĻšāϞ⧋ āĨ¤ āĻĢāϞ⧇ āĻāϟāĻŋ āϏāϰāϞ āĻ›āĻ¨ā§āĻĻāĻŋāϤ āĻ—āϤāĻŋāϤ⧇ āĻ¸ā§āĻĒāĻ¨ā§āĻĻāĻŋāϤ āĻšāϤ⧇ āϞāĻžāĻ—āϞ āĨ¤

(i) āĻŦā§āϞāĻ•āϟāĻŋāϕ⧇ āĻ›ā§‡ā§œā§‡ āĻĻ⧇āĻ“ā§ŸāĻžāϰ āĻĒā§‚āĻ°ā§āĻŦ āĻŽā§āĻšā§āĻ°ā§āϤ⧇ āĻāϰ āωāĻĒāϰ āĻ¸ā§āĻĒā§āϰāĻŋāĻ‚ āĻ•āĻ°ā§āϤ⧃āĻ• āĻĒā§āϰāϝ⧁āĻ•ā§āϤ āĻŦāϞ āĻ•āϤ?

(ii) āĻŦā§āϞāĻ•āϟāĻŋ āĻ›ā§‡ā§œā§‡ āĻĻ⧇āĻ“ā§ŸāĻžāϰ āĻĒāϰ āĻĒāĻ°ā§āϝāĻžā§ŸāĻ•āĻžāϞ āĻ•āϤ?

(iii) āĻ—āϤāĻŋāϰ āĻŦāĻŋāĻ¸ā§āϤāĻžāϰ āĻ•āϤ?

(iv) āĻĻā§‹āϞāĻžā§ŸāĻŽāĻžāύ āĻŦā§āϞāĻ•āϟāĻŋāϰ āϏāĻ°ā§āĻŦāĻžāϧāĻŋāĻ• āĻ—āϤāĻŋāĻŦ⧇āĻ— āĻ•āϤ?

(v) āĻŦā§āϞāĻ•āϟāĻŋāϰ āϏāĻ°ā§āĻŦāĻžāϧāĻŋāĻ• āĻ¤ā§āĻŦāϰāĻŖ āĻ•āϤ?

(vi) āĻŦā§āϞāĻ•āϟāĻŋāϰ āĻŽāĻ§ā§āϝāĻžāĻŦāĻ¸ā§āĻĨāĻžāύ āĻĨ⧇āϕ⧇ āĻ—āϤāĻŋāĻĒāĻĨ⧇āϰ āĻĻāĻŋāϕ⧇ āϝāĻ–āύ āĻ…āĻ°ā§āϧāĻĒāĻĨ āϝāĻžā§Ÿ āϝ⧇ āĻŽā§āĻšā§āĻ°ā§āϤ⧇ āϤāĻžāϰ āĻŦ⧇āĻ—, āĻ¤ā§āĻŦāϰāĻŖ, āĻ¸ā§āĻĨāĻŋāϤāĻŋāĻļāĻ•ā§āϤāĻŋ āĻ“ āĻ—āϤāĻŋāĻļāĻ•ā§āϤāĻŋ āĻŦ⧇āϰ āĻ•āϰ āĨ¤

 

āϏāĻŽāĻžāϧāĻžāύ :

 

(i) F = -kx = -30×(10/100) = -3N                                                          [ans.]

(ii) T = 2Ī€âˆš(m/k) = 2Ī€âˆš(.5/30) = 0.81s                                                [ans.]

(iii) A = 10cm = 0.1m                                                                          [ans.]

(iv) Vm = AΉ = A.(2Ī€/T) = .1×(2Ī€/.81) = 0.77ms-1                   [ans.]

(v) am = Ή2A = (2Ī€/T)2×A = (2Ī€/.81)2×.1 = 6 ms-2                    [ans.]

(vi) V = Ī‰âˆš(A2-x2)

            = (2 Ī€/.81) √(.12-.052) = .67ms-1                                                                       [ans.]

āĻ¤ā§āĻŦāϰāĻŖ, a = Ή2x = (2Ī€/.81)×.05 = 3ms-1                                                    [ans.]

āĻ—āϤāĻŋāĻļāĻ•ā§āϤāĻŋ, T = ÂŊ mv2 = ÂŊ × .5 × .672 = .112J                                          [ans.]

āĻ¸ā§āĻĨāĻŋāϤāĻŋāĻļāĻ•ā§āϤāĻŋ, U = ÂŊ kx2 = ÂŊ × 30 × .052 = 0.038J                           [ans.]

 

 

⧍. āĻāĻ•āϟāĻŋ āϏāϰāϞ āĻĻā§‹āϞāĻ• 2 āĻŽāĻŋāύāĻŋāĻŸā§‡ 60 āĻŦāĻžāϰ āĻĻā§‹āϞāύ āĻĻā§‡ā§Ÿ āĨ¤ āĻĻā§‹āϞāĻ•āϟāĻŋāϰ āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ āύāĻŋāĻ°ā§āϪ⧟ āĻ•āϰ āĨ¤

 

āϏāĻŽāĻžāϧāĻžāύ :

 

T = 2Ī€âˆš(L/g)

⇒ L = (gT2)/(4 Ī€2) = 9.8×(60×260)24Ã—Ī€2

∴ L = 0.993m               [Answer]

 

ā§Š. āϏāϰāϞ āĻ›āĻ¨ā§āĻĻāĻŋāϤ āĻ—āϤāĻŋ āϏāĻŽā§āĻĒāĻ¨ā§āύ āĻāĻ•āϟāĻŋ āĻŦāĻ¸ā§āϤ⧁āϰ āĻŦāĻŋāĻ¸ā§āϤāĻžāϰ 0.01m āĻāĻŦāĻ‚ āĻ•āĻŽā§āĻĒāĻžāĻ‚āĻ• 12Hz āĨ¤ āĻŦāĻ¸ā§āϤ⧁āϟāĻŋāϰ āϏāϰāĻŖ 5×10-3 āĻšāϞ⧇ āĻāϰ āĻ—āϤāĻŋāĻŦ⧇āĻ— āĻ•āϤ?

 

āϏāĻŽāĻžāϧāĻžāύ :

 

V = Ī‰âˆš(A2-x2)                                                  Ή = 2Ī€n

2Ī€Ã—12.012−(5×10−3)2              n = 12Hz

∴ V = 0.654ms-1                                   [Answer]               A = .01m

                                                                                    x = 5×10-3m

 

 

ā§Ē. āϕ⧋āύ āĻāĻ•āϟāĻŋ āϏ⧇āϕ⧇āĻ¨ā§āĻĄ āĻĻā§‹āϞāϕ⧇āϰ āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ āϝāĻĻāĻŋ 1.44 āϗ⧁āĻŖ āĻŦ⧃āĻĻā§āϧāĻŋ āĻ•āϰāĻž āĻšā§Ÿ āϤāĻžāĻšāϞ⧇ āĻāϰ āĻĻā§‹āϞāύāĻ•āĻžāϞ āĻ•āϤ āĻšāĻŦ⧇?

 

āϏāĻŽāĻžāϧāĻžāύ :

 

T1/T2 = √(L1/L2)                                                T1 = 2s

⇒ T2 = T1√(L2/L1)                                            T2 = ?

            = 2√1.44                                              L2 = 1.44L1

∴ T2 = 2.43s                 [Answer]

 

ā§Ģ. āĻāĻ•āϟāĻŋ āϏāϰāϞ āĻĻā§‹āϞāĻ• 0.9s āĻ āĻāĻ•āĻŦāĻžāϰ āϟāĻŋāĻ• āĻļāĻŦā§āĻĻ āĻ•āϰ⧇ āĨ¤ āĻĻā§‹āϞāĻ•āϟāĻŋāϰ āĻ•āĻžāĻ°ā§āϝāĻ•āϰ āĻĻ⧈āĻ°ā§āĻ˜ā§āϝ āĻ•āϤ?

 

āϏāĻŽāĻžāϧāĻžāύ :

 

L = (gT2)/4Ī€2) = {9.8×(.9×2)2}/(4Ī€2)                            T = .9×2s

∴ L = 0.80nm               [Answer]

 

ā§Ŧ. āĻāĻ•āϟāĻŋ āϏ⧇āϕ⧇āĻ¨ā§āĻĄ āĻĻā§‹āϞāĻ• 21.6s āϧ⧀āϰ⧇ āϚāϞ⧇ āĨ¤ āĻ–āύāĻŋāϰ āĻ—āĻ­ā§€āϰāϤāĻž āύāĻŋāĻ°ā§āϪ⧟ āĻ•āϰ āĨ¤ [R = 4000mile]

 

āϏāĻŽāĻžāϧāĻžāύ :

(86400-x)/86400 = √{(R-h)/R}                                                R = 4000mile

⇒ (86400-21.6)/86400 = √{(4000-h)/4000}                            x = 21.6

⇒ h = 1.999 ≈ 2

∴ h = 2 mile                 [Answer]