### Q.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 *x*_{0} and pushed towards the centre with a velocity *v*_{0} at time *t *= 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, *x*_{0}and *v*_{0}. [Hint: Start with the equation *x *= *a *cos (*ω**t**+θ*) and note that the initial velocity is negative.]

**Answer:-**

The displacement rquation for an oscillating mass is given by:

*x* = *A*cos(ω*t* + *θ*)

where,

*A* is the amplitude

*x* is the displacement

*θ *is the phase constant

Velcoity, *v* = *dx*/*dt* = –*A*ωsin(ω*t* + *θ*)

At *t* = 0, *x* = *x*_{0}

x_{0} = *A*cos *θ = x*_{0} ….**(i)**

and, *dx*/*dt* = –*v*_{0} = *A*ωsin*θ*

*A*sin*θ *= *v*_{0}/ω …**(ii)**

Squaring and adding equations **(i)** and **(ii)**, we get: