Physics, asked by percy2004, 11 months ago

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Answered by khanaffan8506
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Explanation:

Differential Equation of SHM and its Solution

Consider a particle of mass (m) executing Simple Harmonic Motion along a path x o x; the mean position at O. Let the speed of the particle be v0 when it is at position p (at a distance no from O)

At t = 0 the particle at P(moving towards right)

At t = t the particle is at Q(at a distance x from O)

With a velocity (v)

Differential Equation of Simple Harmonic Motion

The restoring force F→ at Q is given by

⇒ F→=−Kx→ K – is positive constant

⇒ F→=ma→ a→- acceleration at Q

⇒ ma→=−Kx→

⇒ a→=−(Km)x→

Put, Km=ω2

⇒ ω=Km−−√

⇒ a→=−(Km)m→=−ω2x→ Since, [a→=d2xdt2] d2x→dt2=−ω2x→

d2x/dt2 + ω2x = 0, which is the differential equation for linear simple harmonic motion.

Solutions of Differential Equation of SHM

The differential equation for the Simple harmonic motion has the following solutions:

x=Asinωt (This solution when the particle is in its mean position point (O) in figure (a)

x0=Asinϕ (When the particle is at the position & (not at mean position) in figure (b)

x=Asin(ωt+ϕ) (When the particle at Q at in figure (b) (any time t).

Answered by yuvraj154615
0

Answer:

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