A body of mass m moves according to x = Asinω t , y = Bcosω…

A body of mass $m$ moves according to $x = A\sin\omega t$, $y = B\cos\omega t$, $z = 0$, where $A$, $B$ and $\omega$ are constants. Find its velocity, acceleration and the force acting on it.

Explanation of Method

Force from given equations of motion.

Step 1: Velocity

\[ \vec v = A\omega\cos\omega t\,\hat\imath - B\omega\sin\omega t\,\hat\jmath, \]

\[ v = \omega\sqrt{A^2\cos^2\omega t + B^2\sin^2\omega t}. \]

Step 2: Acceleration

\[ \vec a = -A\omega^2\sin\omega t\,\hat\imath - B\omega^2\cos\omega t\,\hat\jmath = -\omega^2\,\vec r. \]

Step 3: Force

\[\vec F = m\vec a = -m\omega^2\,\vec r\]

The force always points to the origin and is proportional to the distance from it. The path is the ellipse $x^2/A^2 + y^2/B^2 = 1$.

$\vec v$ $\vec F = -m\omega^2\vec r$ $x$ $y$ $A$ $B$ $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$

The body moves clockwise around an ellipse with semi-axes $A$ and $B$. The force always points to the centre and is proportional to the distance from it.

Result

\[\boxed{\,\vec F = m\vec a = -m\omega^2\,\vec r\,}\]

Remark

Check. The original results are correct. Writing the force as $-m\omega^2\vec r$ shows it is a central, spring-like force (a two-dimensional harmonic oscillator).