answer the two questions
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Answers
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).
Answer:
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