Oscillations

Question : 25

Total: 25

A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t=0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0.

Solution:

Let the displacement of the particle at any time t be represented by
x=A‌cos(ωt+ϕ0)...(i)
where A = amplitude, ϕ0= initial phase
If v be the velocity of the particle at time t, Then

[A‌cos(ωt+ϕ0)]=−Aω‌sin(ωt+ϕ0)...(ii)

At t=0,x=x0 and v=v0
By putting t = 0, from (i) and (ii), we get
x0=A‌cos‌ϕ0,
v0=−A‌cos‌ϕ0
⇒v0=−ω√(A‌sin‌ϕ0)2
=−ω√(A2(1−cos2ϕ0) =−ω√A2−A2cos2ϕ0
=−ω√A2−x02...(iii)
Equation (iii) shows that initial velocity is negative. Squaring on both sides of equation (iii), we get
v02=ω(A2−x02) ;
A2−x02=

v02

ω2

⇒A2=x02+

v02

ω2

;
A=√(x02+

v02

ω2

)

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