Question

Hello friends !!

(Q.1) d²x/dt² + 16x = 0 find time period of S.H.M

(Q. 2) If v vs x graph is circle and max speed is 20mm . find Amplitude of S.H.M

Answer 1

Hi

2 answers · Mathematics

Best Answer

Find the general solution by solving the auxiliary equation:

d²x / dt² + 16x = 0

m² + 16 = 0

m² = -16

m = ±4i

y = C₁sin(4t) + C₂cos(4t)

Find the particular solution by solving for the constants:

When t = 0, x = 3

C₂ = 3

y' = 4C₁cos(4t) - 4C₂sin(4t)

When t = 0, dx / dt = 16

4C₁ = 16

C₁ = 4

y = 4sin(4t) + 3cos(4t)

Rewrite this result as a single trigonometric function by comparing coefficients:

4sin(4t) + 3cos(4t) = ksin(4t + α)

4sin(4t) + 3cos(4t) = k[sin(4t)cosα + sinαcos(4t)]

4sin(4t) + 3cos(4t) = (kcosα)sin(4t) + (ksinα)cos(4t)

kcosα = 4

ksinα = 3

(ksinα)² + (kcosα)² = 4² + 3²

k²sin²α + k²cos²α = 25

k²(sin²α + cos²α) = 25

k² = 25

k = 5

ksinα / (kcosα) = ¾

sinα / cosα = ¾

tanα = ¾

α = tanˉ¹(¾)

4sin(4t) + 3cos(4t) = 5sin[4t + tanˉ¹(¾)]

x = 5sin[4t + tanˉ¹(¾)]

Find the maximum values by taking the values of the amplitudes:

x = 5sin[4t + tanˉ¹(¾)]

max(x) = 5 m

dx / dt = 20cos[4t + tanˉ¹(¾)]

max(dx / dt) = 20 m / s

Answer 2

$\bold \pink{hello !!! }$

$\bold \green{solution}$
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Q. 1 )

Solution :- d²x / d²t + w²x = 0 (equation )

comparing d²x/d²t + 16x to equation

Since, se get

w² = 16

w = 4

w = 4

but , w = 2π/t

=> 2π /T= 4

T = π/2 second Answer ✔
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Question :- 2

$\bold\green{solution}$

w = 1 rad/s

Vmax = 20mm /s

WA = 20mm/s

1 * A = 20mm/s

A = 20mm Answer ✔

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