Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $\omega_1$ and $\omega_2$ and have total energies $E_1$ and $E_2$, respectively. The variations of their momenta $p$ with positions $x$ are shown in the figures. If $\frac{a}{b}= n ^2$ and $\frac{ a }{ R }= n$, then the correct equation$(s)$ is(are) $Image$
$(A)$ $E_1 \omega_1=E_2 \omega_2$ $(B)$ $\frac{\omega_2}{\omega_1}=n^2$ $(C)$ $\omega_1 \omega_2= n ^2$ $(D)$ $\frac{E_1}{\omega_1}=\frac{E_2}{\omega_2}$
IIT 2015, Medium
Download our app for free and get started
For first oscillator
$E_1=\frac{1}{2} m \omega_1^2 a^2 \text {, and } p = mv = m \omega_1 a = b \Rightarrow \frac{ a }{ b }=\frac{1}{ m \omega_1} .$ $............(i)$
For second oscillator
$E_2=\frac{1}{2} m \omega_2^2 R^2 \text {, and } m \omega_2=1$ $............(ii)$
$\left(\frac{a}{b}\right)=\frac{\omega_2}{\omega_1}=n^2$
$\frac{E_1}{\omega_1^2 a^2}=\frac{E_2}{\omega_2^2 R^2} \Rightarrow \frac{E_1}{\omega_1}=\frac{E_2}{\omega_2}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A simple pendulum of mass $m$ executes $S.H.M.$ with total energy $E$. If at an instant it is at one of extreme positions, then its linear momentum after a phase shift of $\frac{\pi}{3} \,rad$ will beView Solution
- 2View SolutionThis is the position time graph of a mass on spring. What can you say about the velocity and force at the instant indicated by dashed line ? (positive direction is to the right)
- 3The periodic time of a body executing simple harmonic motion is $3\, sec$. After how much interval from time $t = 0$, its displacement will be half of its amplitude ..... $\sec$View Solution
- 4A linear harmonic oscillator of force constant $2 \times 10^6\,Nm^{-1}$ and amplitude $0.01\, m$ has a total mechanical energy of $160\, J.$ ItsView Solution
- 5In the figure shown, there is friction between the blocks $P$ and $Q$ but the contact between the block $Q$ and lower surface is frictionless. Initially the block $Q$ with block $P$ over it lies at $x=0$, with spring at its natural length. The block $Q$ is pulled to right and then released. As the spring - blocks system undergoes $S.H.M.$ with amplitude $A$, the block $P$ tends to slip over $Q . P$ is more likely to slip atView Solution
- 6In a seconds pendulum, mass of bob is $30\, g$. If it is replaced by $90\, g$ mass. Then its time period will be ... $\sec$View Solution
- 7Two springs with negligible masses and force constant of $K_1 = 200\, Nm^{-1}$ and $K_2 = 160\, Nm^{-1}$ are attached to the block of mass $m = 10\, kg$ as shown in the figure. Initially the block is at rest, at the equilibrium position in which both springs are neither stretched nor compressed. At time $t = 0,$ a sharp impulse of $50\, Ns$ is given to the block with a hammer.View Solution
- 8The function $sin^2\,(\omega t)$ representsView Solution
- 9View SolutionA simple pendulum is taken from the equator to the pole. Its period
- 10Length of a simple pendulum is $l$ and its maximum angular displacement is $\theta$, then its maximum $K.E.$ isView Solution