A mass attached to a spring is free to oscillate, with angular velocity w, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity υₒ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, xₒ and υₒ. [Hint : Start with the equation x = α cos (ωt+θ) and note that the initial velocity is negative.]
The displacement of the mass attached to the spring is given as,
x = α cos ( ω t + θ )
Where, α is the amplitude of the oscillation and ω is the angular acceleration of the motion.
Initial condition is t = 0 , so the displacement is x = x 0 .
By substituting the given values in the above equation, we get
x 0 = α cos θ (1)
The derivative of displacement is the velocity of the body.
v = d x d t = − α ω sin ( ω t + θ )
Initial condition is t = 0 , so the velocity is v = − v 0 .
By substituting the given values in the above equation, we get
− v 0 = − α ω sin ( θ ) α sin θ = v 0 ω (2)
By squaring and adding equations (1) and (2), we get
α 2 ( sin 2 θ + cos 2 θ ) = x 0 2 + v 0 2 ω 2 α = x 0 2 + v 0 2 ω 2
Thus, the amplitude of the resultant oscillation is x 0 2 + v 0 2 ω 2 .
The displacement of the mass attached to the spring is given as,
Where,
Initial condition is
By substituting the given values in the above equation, we get
The derivative of displacement is the velocity of the body.
Initial condition is
By substituting the given values in the above equation, we get
By squaring and adding equations (1) and (2), we get
Thus, the amplitude of the resultant oscillation is
