CBSE (Science) Class 11CBSE
Share

# A Mass Attached to a Spring is Free to Oscillate, with Angular Velocity ω, in a Horizontal Plane Without Friction Or Dampinng Determine the Amplitude of the Resulting Oscillations in Terms of the Parameters ω - CBSE (Science) Class 11 - Physics

ConceptSome Systems Executing Simple Harmonic Motion

#### Question

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 = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. [Hint: Start with the equation acos (ωt) and note that the initial velocity is negative.]

#### Solution 1

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)

#### Solution 2

x = alpha cos (omegat + theta)

v = (dx)/(dt) = -aomega sin(omegat = theta)

When t = 0, x = x_0 and dx/dt = -v_0

x= 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)

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [1]

Solution A Mass Attached to a Spring is Free to Oscillate, with Angular Velocity ω, in a Horizontal Plane Without Friction Or Dampinng Determine the Amplitude of the Resulting Oscillations in Terms of the Parameters ω Concept: Some Systems Executing Simple Harmonic Motion.
S