Write down the Equations of Motion of Particle moving under Central Force.
ANSWER : When a particle moves under a central force, it's motion can be described by the following two equations :
Radial Equation :
\[m\frac{{d^2r}}{{dt^2}} = -F(r) + m \cdot r \left(\frac{{d^2\theta}}{{dt^2}}\right)^2\]
In this equation, \(m\) represents the mass of the particle. The term \(\frac{{d^2r}}{{dt^2}}\) represents the second derivative of the radial distance \(r\) with respect to time \(t\), which gives the acceleration along the radial direction. The force acting on the particle is denoted by \(F(r)\), which is a function of the radial distance \(r\). The term \(r \left(\frac{{d^2\theta}}{{dt^2}}\right)^2\) represents the square of the second derivative of the angular displacement \(\theta\) with respect to time \(t\), which is the centripetal acceleration due to the rotation of the particle around the central point.
Angular Equation :
\[m \cdot r^2 \frac{{d^2\theta}}{{dt^2}} = r \cdot F(\theta)\]
Here, \(r^2\) represents the square of the radial distance \(r\). The term \(\frac{{d^2\theta}}{{dt^2}}\) represents the second derivative of the angular displacement \(\theta\) with respect to time \(t\), which gives the angular acceleration of the particle. \(F(\theta)\) represents the angular force acting on the particle, which depends on the angular displacement \(\theta\).
These equations describe the motion of a particle under a central force, where the force acting on the particle is solely dependent on the distance from the center and/or the angular position around the center. Solving these equations helps us understand the trajectory and behavior of the particle in such a system.
Also Read : What is Driven Simple Harmonic Oscillator?
Also Read : Give the expression for Time Period of Compound Pendulum.
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