A graph of the square of the velocity against the square of the acceleration of a given simple harmonic motion is
Diffcult
Download our app for free and get started
$x=\mathrm{A} \sin \omega \mathrm{t}$
$\mathrm{v}=\frac{\mathrm{d} \mathrm{x}}{\mathrm{dt}}=\mathrm{A} \omega \cos \omega \mathrm{t}=\omega \sqrt{\mathrm{A}^{2}-\mathrm{x}^{2}}$
$\mathrm{a}=\frac{\mathrm{d} \mathrm{v}}{\mathrm{dt}}=-\mathrm{A} \omega^{2} \sin \omega \mathrm{t}$
$\mathrm{a}=\frac{\mathrm{d} \mathrm{v}}{\mathrm{dt}}=-\omega^{2} \mathrm{x}$
But $x=-\frac{a}{\omega^{2}}$
$\therefore \quad v=\omega \sqrt{A^{2}-\frac{a^{2}}{\omega^{4}}}$
or $v^{2}=\omega^{2}\left(A^{2}-\frac{a^{2}}{\omega^{4}}\right)$
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 length $l$ is made to oscillate with an amplitude of $45$ degrees. The acceleration due to gravity is $g$. Let $T_0=2 \pi \sqrt{l / g}$. The time period of oscillation of this pendulum will beView Solution
- 2A block with mass $M$ is connected by a massless spring with stiffiess constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: ($i$) when the block is at $x_0$; and ($ii$) when the block is at $x=x_0+A$. In both the cases, a perticle with mass $m$ is placed on the mass $M$ ?View Solution
($A$) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged
($B$) The final time period of oscillation in both the cases is same
($C$) The total energy decreases in both the cases
($D$) The instantaneous speed at $x_0$ of the combined masses decreases in both the cases
- 3View SolutionWhen a particle executes simple Harmonic motion, the nature of graph of velocity as function of displacement will be.
- 4The displacement of an object attached to a spring and executing simple harmonic motion is given by $ x= 2 \times 10^{-9}$ $ cos$ $\;\pi t\left( m \right)$ .The time at which the maximum speed first occurs isView Solution
- 5View SolutionResonance is an example of
- 6A block of mass $200\, g$ executing $SHM$ under the influence of a spring of spring constant $K=90\, N\,m^{-1}$ and a damping constant $b=40\, g\,s^{-1}$. The time elapsed for its amplitude to drop to half of its initial value is ...... $s$ (Given $ln\,\frac{1}{2} = -0.693$)View Solution
- 7The displacement of a particle undergoing $SHM$ of time period $T$ is given by $x(t) = x_m\,cos\, (\omega t + \phi )$. The particle is at $x = -x_m$ at time $t = 0$. The particle is at $x = + x_m$ whenView Solution
- 8A particle executing simple harmonic motion has an amplitude of $6\, cm$. Its acceleration at a distance of $2 \,cm$ from the mean position is $8\,cm/{s^2}$. The maximum speed of the particle is ... $ cm/s$View Solution
- 9The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If length of spring is cut to one third then the same mass oscillates with frequency $v_2$, thenView Solution
- 10Find the ratio of time periods of two identical springs if they are first joined in series $\&$ then in parallel $\&$ a mass $m$ is suspended from them :View Solution