A mass attached to a spring is free to oscillate, with angular velocity $\omega,$ in a hortzontal plane without friction or damping. It is pulled to a distance $x_{0}$ and pushed towards the centre with a velocity $v_{ o }$ at time $t=0 .$ Determine the amplitude of the resulting oscillations in terms of the parameters $\omega, x_{0}$ and $v_{ o } .$ [Hint: Start with the equation $x=a \cos (\omega t+\theta)$ and note that the initial velocity is negative.]
A mass attached to a spring is free to oscillate, with angular velocity $\omega,$ in a hortzontal plane without friction or damping. It is pulled to a distance $x_{0}$ and pushed towards the centre with a velocity $v_{ o }$ at time $t=0 .$ Determine the amplitude of the resulting oscillations in terms of the parameters $\omega, x_{0}$ and $v_{ o } .$ [Hint: Start with the equation $x=a \cos (\omega t+\theta)$ and note that the initial velocity is negative.]
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 $\theta$
is the phase constant
Velocity, $v=\frac{d x}{d t}=-A \omega \sin (\omega t+\theta)$
At $t=0, x=x_{0}$
$A \cos \theta=x_{0} \ldots(i)$
And, $\frac{d x}{d t}=-v_{0}=A \omega \sin \theta$
$A \sin \theta=\frac{v_{0}}{\omega} \ldots(i i)$
Squaring and adding equations ( $i$ ) and ($ ii $), we get
$A^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=x_{0}^{2}+\left(\frac{v_{0}^{2}}{\omega^{2}}\right)$
$\therefore A=\sqrt{x_{0}^{2}+\left(\frac{v_{0}}{\omega}\right)^{2}}$
Hence, the amplitude of the resulting oscillation is $\sqrt{x_{0}^{2}+\left(\frac{v_{0}}{\omega}\right)^{2}}$
Similar Questions
A $5\; kg$ collar is attached to a spring of spring constant $500\;N m ^{-1} .$ It slides without friction over a hortzontal rod. The collar is displaced from its equilibrium position by $10.0\; cm$ and released. Calculate
$(a)$ the period of oscillation.
$(b)$ the maximum speed and
$(c)$ maximum acceleration of the collar.
A $5\; kg$ collar is attached to a spring of spring constant $500\;N m ^{-1} .$ It slides without friction over a hortzontal rod. The collar is displaced from its equilibrium position by $10.0\; cm$ and released. Calculate
$(a)$ the period of oscillation.
$(b)$ the maximum speed and
$(c)$ maximum acceleration of the collar.
A spring is stretched by $5 \,\mathrm{~cm}$ by a force $10 \,\mathrm{~N}$. The time period of the oscillations when a mass of $2 \,\mathrm{~kg}$ is suspended by it is :(in $s$)
A spring is stretched by $5 \,\mathrm{~cm}$ by a force $10 \,\mathrm{~N}$. The time period of the oscillations when a mass of $2 \,\mathrm{~kg}$ is suspended by it is :(in $s$)
- [NEET 2021]
A mass $m$ attached to a spring oscillates with a period of $3\,s$. If the mass is increased by $1\,kg$ the period increases by $1\,s$. The initial mass $m$ is
A mass $m$ attached to a spring oscillates with a period of $3\,s$. If the mass is increased by $1\,kg$ the period increases by $1\,s$. The initial mass $m$ is
The springs shown are identical. When $A = 4kg$, the elongation of spring is $1\, cm$. If $B = 6\,kg$, the elongation produced by it is ..... $ cm$
The springs shown are identical. When $A = 4kg$, the elongation of spring is $1\, cm$. If $B = 6\,kg$, the elongation produced by it is ..... $ cm$
How the period of oscillation depend on the mass of block attached to the end of spring ?
How the period of oscillation depend on the mass of block attached to the end of spring ?