A simple harmonic oscillator consists of a mass attached to a spring. The mass moves back and forth due to the restoring force exerted by the spring. The parameters given for this system are the mass \(m\) kg and the spring constant \(k\) Nm-1.
The equation for a simple harmonic oscillator is given by:
\(F = -kx = ma\)
The acceleration \(a\) can be linked to velocity \(v\) and position \(x\) using the energy conservation principle:
\(\frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \text{constant}\)
The equation above signifies that the sum of kinetic and potential energy remains constant, representing an ellipse in the \(x-v\) phase space.
For the graph of \(v\) as a function of \(x\) to be a circle, the equation should be of the form where the coefficient of \(x^2\) and \(v^2\) are equal. Thus:
\(\frac{1}{2}mv^2 + \frac{1}{2}kx^2 = E \quad \Rightarrow \quad mv^2 = kx^2\)
For the coefficients of \(v^2\) and \(x^2\) to be equal, we require:
\(m = k\)
Therefore, for the graph to be a circle, the relationship \(k = m\) must hold true.
Hence, the correct option is:
$k=m$
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Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
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- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- The equation of a transverse wave is $y = y_0 \sin 2\pi(ft - \frac{x}{\lambda})$. If the maximum particle velocity be four times that of wave velocity then
- The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}~\text{s}$, the value of $x$ is ________.
