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प्रश्न
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. [Hint: Start with the equation x = acos (ωt+θ) and note that the initial velocity is negative.]
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उत्तर १
The displacement equation for an oscillating mass is given by:
x = `A cos (omega t + theta)`
Where,
A is the amplitude
x is the displacement
θ is the phase constant
Velocity, `v = (dx)/(dt) = -Aomega sin(omega t + theta)`
At t = 0, x = x0
x0 = Acosθ = x0 … (i)
And, `(dx)/(dt) = -v_0 = Aomegasin theta`
`A sin theta = v_0/omega` ...(ii)
Squaring and adding equations (i) and (ii), we get:
`A^2(cos^2 theta + sin^2 theta)= x_0^2 + ((v_0^2)/(omega^2))`
`:. A= sqrt(x_0^2 + (v_0/omega)^2)`
Hence, the amplitude of the resulting oscillation is `sqrt(x_0^2 + (v_0/omega)^2)`
उत्तर २
x = `alpha cos (omegat + theta)`
`v = (dx)/(dt) = -aomega sin(omegat = theta)`
When t = 0, x = `x_0` and `dx/dt = -v_0`
x0 = `a cos theta`
and `-v_0 = -a omega sin theta` or a `sin theta = v_0/omega`
Squaring and adding (i) and (ii) we get
`a^2(cos^2 theta + sin^2 theta) = x_0^2 + (v_0^2)/(omega^2)`
or `a = sqrt(x_0^2 + v_0^2/omega^2)`
संबंधित प्रश्न
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- increase
- remain same
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- first increase and then decrease.
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(A) `pi/2rad`
(B) π rad
(C) 2π rad
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(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?
Answer the following questions:
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Answer the following questions:
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- Force acting is directly proportional to displacement from the mean position and opposite to it.
- Motion is periodic.
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- The velocity is periodic.
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(In the frame of the trolley)
A particle at the end of a spring executes simple harmonic motion with a period t1, while the corresponding period for another spring is t2. If the period of oscillation with the two springs in series is T, then ______.
If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is `x/2` times its original time period. Then the value of x is ______.
