Question

A particle of mass m oscillates along x-axis according to equation x = asinωt. The nature of the graph between momentum and displacement of the particle is

• A. A straight line
• B. A circle
• C. An ellipse
• D. A parabola

Answer 1

Answer: (C)

An ellipse

Answer 2

The displacement of the particle is given by $x = a \sin(\omega t)$. The velocity is the time derivative of displacement: $v = \frac{dx}{dt} = a \omega \cos(\omega t)$. Momentum is given by $p = mv$, so $p = m(a \omega \cos(\omega t)) = ma \omega \cos(\omega t)$. From the displacement equation, we can express $\sin(\omega t)$ as $\frac{x}{a}$. From the momentum equation, we can express $\cos(\omega t)$ as $\frac{p}{ma \omega}$. Using the fundamental trigonometric identity $\sin^2(\theta) + \cos^2(\theta) = 1$, we substitute $\theta = \omega t$: $\sin^2(\omega t) + \cos^2(\omega t) = 1$ Substituting the expressions for $\sin(\omega t)$ and $\cos(\omega t)$: $\left(\frac{x}{a}\right)^2 + \left(\frac{p}{ma \omega}\right)^2 = 1$ This equation can be rewritten as: $\frac{x^2}{a^2} + \frac{p^2}{(ma \omega)^2} = 1$ This is the standard form of the equation of an ellipse centered at the origin $(0,0)$ in the $x-p$ plane. The semi-major/minor axes are $a$ along the x-axis and $ma\omega$ along the p-axis.